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{{For|इन समीकरणों में प्रयुक्त गणित अंकन की सूची|गणितीय अंकन}}
{{Thermodynamics|cTopic=[[Thermodynamic equations|Equations]]}}
{{main|थर्मोडायनामिक गुणों की सूची}}

यह लेख [[ऊष्मप्रवैगिकी]] में सामान्य [[समीकरण]] और [[भौतिक मात्रा]] का सारांश है (अधिक विस्तार के लिए ऊष्मप्रवैगिकी समीकरण देखें)।

== परिभाषाएँ ==

{{Main article|ऊष्मागतिक गुणों की सूची|ऊष्मागतिक विभव|मुक्त एन्ट्रापी|परिभाषित समीकरण (भौतिक रसायन विज्ञान)}}

नीचे दी गई कई परिभाषाएँ [[रासायनिक प्रतिक्रिया]]ओं के ऊष्मप्रवैगिकी में भी उपयोग की जाती हैं।

=== सामान्य मूल मात्रा ===

{| class="wikitable"
|-
! scope="col" width="200" | Quantity (Common Name/s) 
! scope="col" width="125" | (Common) Symbol/s 
! scope="col" width="125" | SI Units 
! scope="col" width="100" | Dimension
|-
!Number of molecules
| ''N''
| dimensionless
| dimensionless
|-
!Number of moles
|''n''
| mol
| [N]
|-
![[Temperature]]
| ''T''
| K
| [Θ]
|-
![[Heat|Heat Energy
]]
| ''Q, q''
| J
| [M][L]<sup>2</sup>[T]<sup>−2</sup>
|-
![[Latent heat]]
| ''Q<sub>L</sub>''
| J
| [M][L]<sup>2</sup>[T]<sup>−2</sup>
|-
|}


=== सामान्य व्युत्पन्न मात्रा ===

{| class="wikitable"
|-b 

! scope="col" width="200" | Quantity (Common Name/s) 
! scope="col" width="125" | (Common) Symbol/s 
! scope="col" width="200" | Defining Equation 
! scope="col" width="125" | SI Units 
! scope="col" width="100" | Dimension
|-
![[Thermodynamic beta]], Inverse temperature 
|| ''β''
||<math> \beta = 1/k_B T \,\!</math>
|| J<sup>−1</sup>
|| [T]<sup>2</sup>[M]<sup>−1</sup>[L]<sup>−2</sup>
|-
![[Thermodynamic temperature]]
| ''τ''
|<math> \tau = k_B T \,\!</math>
<math> \tau = k_B \left (\partial U/\partial S \right )_{N} \,\!</math>
<math> 1/\tau = 1/k_B \left (\partial S/\partial U \right )_{N} \,\!</math>
| J
| [M] [L]<sup>2</sup> [T]<sup>−2</sup>
|-
![[Entropy]]
| ''S''
| <math>S = -k_B\sum_i p_i\ln p_i</math>
<math> S = -\left (\partial F/\partial T \right )_{V} \,\!</math>   ,
<math> S = -\left (\partial G/\partial T \right )_{N,P} \,\!</math>
| J K<sup>−1</sup>
| [M][L]<sup>2</sup>[T]<sup>−2</sup> [Θ]<sup>−1</sup>
|-
![[Pressure]]
| ''P'' 
| <math> P = - \left (\partial F/\partial V \right )_{T,N} \,\!</math>
<math> P = - \left (\partial U/\partial V \right )_{S,N} \,\!</math>
| Pa
| M  L<sup>−1</sup>T<sup>−2</sup>
|-
![[Internal Energy]]
| ''U''
|<math>U = \sum_i  E_i \!</math>
| J
| [M][L]<sup>2</sup>[T]<sup>−2</sup>
|-
![[Enthalpy]]
| ''H''
|<math> H = U+pV\,\!</math>
| J
| [M][L]<sup>2</sup>[T]<sup>−2</sup>
|-
![[Partition function (statistical mechanics)|Partition Function]] 
| ''Z''
|
| dimensionless
| dimensionless
|-
![[Gibbs free energy]]
| ''G''
|<math> G = H - TS \,\!</math>
| J
| [M][L]<sup>2</sup>[T]<sup>−2</sup>
|-
![[Chemical potential]] (of 
component ''i'' in a mixture)
| ''μ<sub>i</sub>''
|<math> \mu_i = \left (\partial U/\partial N_i \right )_{N_{j \neq i}, S, V } \,\!</math>
<math> \mu_i = \left (\partial F/\partial N_i \right )_{T, V } \,\!</math>, where F is not proportional to N because μ<sub>i</sub> depends on pressure.
<math> \mu_i = \left (\partial G/\partial N_i \right )_{T, P } \,\!</math>, where G is proportional to N (as long as the molar ratio composition of the system remains the same) because μ<sub>i</sub> depends only on temperature and pressure and composition.
<math> \mu_i/\tau = -1/k_B \left (\partial S/\partial N_i \right )_{U,V} \,\!</math>
| J
| [M][L]<sup>2</sup>[T]<sup>−2</sup>
|-
![[Helmholtz free energy]]
| ''A, F''
|<math> F = U - TS \,\!</math>
| J
| [M][L]<sup>2</sup>[T]<sup>−2</sup>
|-
![[Grand potential|Landau potential]], Landau Free Energy, [[Grand potential]]
| ''Ω'', ''Φ<sub>G</sub>''
|<math> \Omega = U - TS - \mu N\,\!</math>
| J
| [M][L]<sup>2</sup>[T]<sup>−2</sup>
|-
!Massieu Potential, Helmholtz [[free entropy]]
| ''Φ''
|<math> \Phi = S - U/T \,\!</math>
| J K<sup>−1</sup>
| [M][L]<sup>2</sup>[T]<sup>−2</sup> [Θ]<sup>−1</sup>
|-
!Planck potential, Gibbs [[free entropy]]
| ''Ξ''
|<math> \Xi = \Phi - pV/T \,\!</math>
| J K<sup>−1</sup>
| [M][L]<sup>2</sup>[T]<sup>−2</sup> [Θ]<sup>−1</sup>
|-
|}


=== पदार्थ के ऊष्मीय गुण ===

{{Main articles|ऊष्मा धारिता|तापीय प्रसार}}

{| class="wikitable"
|-
! scope="col" width="100" | Quantity (common name/s) 
! scope="col" width="100" | (Common) symbol/s 
! scope="col" width="300" | Defining equation 
! scope="col" width="125" | SI units 
! scope="col" width="100" | Dimension
|-
!General heat/thermal capacity 
|| ''C''
||<math> C = \partial Q/\partial T\,\!</math>
|| J K <sup>−1</sup>
|| [M][L]<sup>2</sup>[T]<sup>−2</sup> [Θ]<sup>−1</sup>
|-
!Heat capacity (isobaric)
|| ''C<sub>p</sub>''
||<math> C_{p} = \partial H/\partial T\,\!</math>
|| J K <sup>−1</sup>
|| [M][L]<sup>2</sup>[T]<sup>−2</sup> [Θ]<sup>−1</sup>
|-
!Specific heat capacity (isobaric)
|| ''C<sub>mp</sub>''
||<math> C_{mp} = \partial^2 Q/\partial m \partial T \,\!</math>
|| J kg<sup>−1</sup> K<sup>−1</sup> 
| [L]<sup>2</sup>[T]<sup>−2</sup> [Θ]<sup>−1</sup>
|-
!Molar specific heat capacity (isobaric)
|| ''C<sub>np</sub>''
||<math>C_{np} = \partial^2 Q/\partial n \partial T \,\!</math>
|| J K <sup>−1</sup> mol<sup>−1</sup>
|| [M][L]<sup>2</sup>[T]<sup>−2</sup> [Θ]<sup>−1</sup> [N]<sup>−1</sup>
|-
!Heat capacity (isochoric/volumetric)
|| ''C<sub>V</sub>''
||<math> C_{V} = \partial U/\partial T \,\!</math>
|| J K <sup>−1</sup>
|| [M][L]<sup>2</sup>[T]<sup>−2</sup> [Θ]<sup>−1</sup>
|-
!Specific heat capacity (isochoric)
|| ''C<sub>mV</sub>''
||<math> C_{mV} = \partial^2 Q/\partial m \partial T \,\!</math>
|| J kg<sup>−1</sup> K<sup>−1</sup>
|| [L]<sup>2</sup>[T]<sup>−2</sup> [Θ]<sup>−1</sup>
|-
!Molar specific heat capacity (isochoric)
|| ''C<sub>nV</sub>''
||<math> C_{nV} = \partial^2 Q/\partial n \partial T \,\!</math>
|| J K <sup>−1</sup> mol<sup>−1</sup>
|| [M][L]<sup>2</sup>[T]<sup>−2</sup> [Θ]<sup>−1</sup> [N]<sup>−1</sup>
|-
!Specific latent heat
|| ''L''
||<math>L = \partial Q/ \partial m \,\!</math>
|| J kg<sup>−1</sup>
|| [L]<sup>2</sup>[T]<sup>−2</sup>
|-
!Ratio of isobaric to isochoric heat capacity, [[heat capacity ratio]], adiabatic index
|| ''γ''
||<math>\gamma = C_p/C_V = c_p/c_V = C_{mp}/C_{mV} \,\!</math>
|| dimensionless
|| dimensionless
|-
|}


===थर्मल ट्रांसफर===

{{Main|ऊष्मीय चालकता}}

{| class="wikitable"
|-
! scope="col" width="100" | Quantity (common name/s) 
! scope="col" width="100" | (Common) symbol/s 
! scope="col" width="300" | Defining equation 
! scope="col" width="125" | SI units 
! scope="col" width="100" | Dimension
|-
![[Temperature gradient]]
|| No standard symbol 
||<math> \nabla T \,\!</math>
|| K m<sup>−1</sup>
|| [Θ][L]<sup>−1</sup>
|-
!Thermal conduction rate, thermal current, thermal/[[heat flux]], thermal power transfer
|| ''P''
||<math>P = \mathrm{d} Q/\mathrm{d} t \,\!</math>
|| W = J s<sup>−1</sup>
|| [M] [L]<sup>2</sup> [T]<sup>−3</sup>
|-
!Thermal intensity
|| ''I''
||<math>I = \mathrm{d} P/\mathrm{d} A </math>
|| W m<sup>−2</sup>
|| [M] [T]<sup>−3</sup>
|-
!Thermal/heat flux density (vector analogue of thermal intensity above)
|| '''q'''
||<math>Q = \iint \mathbf{q} \cdot \mathrm{d}\mathbf{S}\mathrm{d} t \,\!</math>
|| W m<sup>−2</sup>
|| [M] [T]<sup>−3</sup>
|-
|}


== समीकरण ==

इस लेख के समीकरणों को विषय द्वारा वर्गीकृत किया गया है।

=== ऊष्मागतिक प्रक्रियाएं ===

{| class="wikitable"
|-
! scope="col" width="100" | Physical situation
! scope="col" width="10" | Equations
|-
!Isentropic process (adiabatic and reversible)
|<math>Q = 0, \quad \Delta U = -W\,\!</math>

For an ideal gas<br />
<math>p_1 V_1^{\gamma} = p_2 V_2^{\gamma}\,\!</math><br />
<math>T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1} \,\!</math><br />
<math>p_1^{1-\gamma} T_1^{\gamma} = p_2^{1 - \gamma} T_2^{\gamma} \,\!</math>
|-
!Isothermal process
|<math>\Delta U = 0, \quad W =  Q \,\!</math>

For an ideal gas<br />
<math>W=kTN \ln(V_2/V_1)\,\!</math>
<math>W=nRT \ln(V_2/V_1)\,\!</math>
|-
!Isobaric process
|''p''<sub>1</sub> = ''p''<sub>2</sub>, ''p'' = constant <br />
<math>W = p \Delta V, \quad Q = \Delta U + p \delta V\,\!</math>
|-
!Isochoric process
|''V''<sub>1</sub> = ''V''<sub>2</sub>, ''V'' = constant <br />
<math>W = 0, \quad Q = \Delta U\,\!</math>
|-
!Free expansion
|<math>\Delta U = 0\,\!</math>
|-
!Work done by an expanding gas
|Process<br />
<math> W = \int_{V_1}^{V_2} p \mathrm{d}V \,\!</math>

Net Work Done in Cyclic Processes<br />
<math> W = \oint_\mathrm{cycle} p \mathrm{d}V = \oint_\mathrm{cycle}\Delta Q \,\!</math>
|-
|}


===गतिज सिद्धांत===

{| class="wikitable"
|+Ideal gas equations
! scope="col" width="100" | Physical situation
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equations
|-
! Ideal gas law
| {{plainlist}}
*''p'' = pressure
*''V'' = volume of container
*''T'' = temperature 
*''n'' = number of moles
*''R'' = [[Gas constant]]
*''N'' = number of molecules
*''k'' = [[Boltzmann constant|Boltzmann's constant]]
{{endplainlist}}
|<math>pV = nRT = kTN\,\!</math><br />
<math>\frac{p_1 V_1}{p_2 V_2} = \frac{n_1 T_1}{n_2 T_2} = \frac{N_1 T_1}{N_2 T_2} \,\!</math>
|-
! Pressure of an ideal gas
| {{plainlist}}
*''m'' = mass of ''one'' molecule
*''M<sub>m</sub>'' = molar mass
{{endplainlist}}
| <math>p = \frac{Nm \langle v^2 \rangle}{3V} = \frac{nM_m \langle v^2 \rangle}{3V} = \frac{1}{3}\rho \langle v^2 \rangle \,\!</math>
|-
|}


==== आदर्श गैस ====

{| class="wikitable sortable mw-collapsible"
|+
|-
! Quantity
! General Equation
! Isobaric<br />Δ''p'' = 0
! Isochoric<br />Δ''V'' = 0
! Isothermal<br />Δ''T'' = 0
! Adiabatic<br /><math>Q=0</math>
|-
! Work <br /> ''W''
| align="center" | <math> \delta W = -p dV\;</math>
| align="center" | <math>-p\Delta V\;</math>
| align="center" | <math>0\;</math>
| align="center" | <math>-nRT\ln\frac{V_2}{V_1}\;</math>
<math>-nRT\ln\frac{P_1}{P_2}\;</math>
| align="center" | <math>\frac{PV^\gamma (V_f^{1-\gamma} - V_i^{1-\gamma}) } {1-\gamma} = C_V \left(T_2 - T_1 \right)</math>
|-
! Heat Capacity <br /> ''C''
| align="center" | (as for real gas)
| align="center" | <math>C_p = \frac{5}{2}nR\;</math><br>(for monatomic ideal gas) 
<math>C_p = \frac{7}{2}nR \;</math><br>(for diatomic ideal gas)
| align="center" | <math>C_V = \frac{3}{2}nR \;</math><br>(for monatomic ideal gas) 
<math>C_V = \frac{5}{2}nR \;</math><br>(for diatomic ideal gas)
||
||
|-
! Internal Energy <br /> Δ''U'' 
| align="center" | <math>\Delta U = C_V \Delta T\;</math>
| align="center" | <math>Q + W\;</math><br><br><math>Q_p - p\Delta V\;</math>
| align="center" | <math>Q\;</math><br><br><math>C_V\left ( T_2-T_1 \right )\;</math>
| align="center" | <math>0\;</math><br><br><math>Q=-W\;</math>
| align="center" | <math>W\;</math><br><br><math>C_V\left ( T_2-T_1 \right )\;</math>
|-
! Enthalpy <br /> Δ''H''
| align="center" | <math>H=U+pV\;</math>
| align="center" | <math>C_p\left ( T_2-T_1 \right )\;</math>
| align="center" | <math>Q_V+V\Delta p\;</math>
| align="center" | <math>0\;</math>
| align="center" | <math>C_p\left ( T_2-T_1 \right )\;</math>
|-
! Entropy <br /> Δ''s''
| align="center" | <math>\Delta S = C_V \ln{T_2 \over T_1} + nR \ln{V_2 \over V_1}</math><br><math>\Delta S = C_p \ln{T_2 \over T_1} - nR \ln{p_2 \over p_1}</math><ref>Keenan, ''Thermodynamics'', Wiley, New York, 1947</ref>
| align="center" | <math>C_p\ln\frac{T_2}{T_1}\;</math>
| align="center" | <math>C_V\ln\frac{T_2}{T_1}\;</math>
| align="center" | <math>nR\ln\frac{V_2}{V_1}\;</math><br><math>\frac{Q}{T}\;</math>
| align="center" | <math>C_p\ln\frac{V_2}{V_1}+C_V\ln\frac{p_2}{p_1}=0\;</math>
|-
! Constant
| <math>\;</math>
| align="center" | <math>\frac{V}{T}\;</math>
| align="center" | <math>\frac{p}{T}\;</math>
| align="center" | <math>p V\;</math>
| align="center" | <math>p V^\gamma\;</math>
|}


=== एंट्रॉपी ===

*<math> S = k_\mathrm{B} \ln \Omega </math>, जहां '''''k'''''<sub>B</sub> बोल्ट्ज़मैन स्थिरांक है, और Ω [[चरण स्थान]] में [[Index.php?title=स्थूल अवस्था|स्थूल अवस्था]] की मात्रा को दर्शाता है या अन्यथा ऊष्मागतिक संभाव्यता कहा जाता है।
*<math> dS = \frac{\delta Q}{T} </math>, केवल प्रतिवर्ती प्रक्रियाओं के लिए

=== सांख्यिकीय भौतिकी ===

नीचे एक आदर्श गैस के लिए मैक्सवेल-बोल्ट्ज़मैन वितरण से उपयोगी परिणाम और एंट्रॉपी मात्रा के निहितार्थ हैं। वितरण आदर्श गैस बनाने वाले परमाणुओं या अणुओं के लिए मान्य है।

{| class="wikitable"
|-
! scope="col" width="100" | Physical situation
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equations
|-
!Maxwell–Boltzmann distribution
| {{plainlist}}
*''v'' = velocity of atom/molecule,
*''m'' = mass of each molecule (all molecules are identical in kinetic theory),
*''γ''(''p'') = Lorentz factor as function of momentum (see below)
*Ratio of thermal to rest mass-energy of each molecule:<math>\theta = k_B T/mc^2 \,\!</math> 
{{endplainlist}}

''K''<sub>2</sub> is the Modified [[Bessel function]] of the second kind.
|Non-relativistic speeds<br />
<math>P\left ( v \right )=4\pi\left ( \frac{m}{2\pi k_B T} \right )^{3/2} v^2 e^{-mv^2/2 k_B T} \,\!</math>

Relativistic speeds (Maxwell-Jüttner distribution)<br />
<math> f(p) = \frac{1}{4 \pi m^3 c^3 \theta K_2(1/\theta)} e^{-\gamma(p)/\theta}</math>
|-
!Entropy [[logarithmic scale|Logarithm]] of the [[density of states]]
|{{plainlist}}
* ''P<sub>i</sub>'' = probability of system in microstate ''i''
* Ω = total number of microstates
{{endplainlist}}
|<math>S = - k_B\sum_i P_i \ln P_i = k_\mathrm{B}\ln \Omega\,\!</math>

where:<br />
<math>P_i = 1/\Omega\,\!</math>
|-
!Entropy change
|
|<math>\Delta S = \int_{Q_1}^{Q_2} \frac{\mathrm{d}Q}{T} \,\!</math><br />
<math>\Delta S = k_B N \ln\frac{V_2}{V_1} + N C_V \ln\frac{T_2}{T_1} \,\!</math>
|-
!Entropic force
|
|<math>\mathbf{F}_\mathrm{S} = -T \nabla S \,\!</math>
|-
!Equipartition theorem
| ''d<sub>f</sub>'' = degree of freedom
| Average kinetic energy per degree of freedom
<math> \langle E_\mathrm{k} \rangle = \frac{1}{2}kT\,\!</math>

Internal energy
<math> U = d_f \langle E_\mathrm{k} \rangle = \frac{d_f}{2}kT\,\!</math>
|- 
|}
गैर-सापेक्षवादी मैक्सवेल-बोल्ट्जमान वितरण के परिणाम नीचे दिए गए हैं।

{| class="wikitable"
|-
! scope="col" width="100" | Physical situation
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equations
|-
!Mean speed
|
|<math> \langle v \rangle = \sqrt{\frac{8 k_B T}{\pi m}}\,\!</math>
|-
!Root mean square speed
|
| <math> v_\mathrm{rms} = \sqrt{\langle v^2 \rangle} = \sqrt{\frac{3k_B T}{m}} \,\!</math>
|-
!Modal speed
|
|<math> v_\mathrm{mode} = \sqrt{\frac{2k_B T}{m}}\,\!</math>
|-
![[Mean free path]]
|{{plainlist}}
*''σ'' = Effective cross-section
*''n'' = Volume density of number of target particles
*''{{ell}}'' = Mean free path 
{{endplainlist}}
|<math>\ell = 1/\sqrt{2} n \sigma \,\!</math>
|-
|}


=== अर्ध-स्थैतिक और प्रतिवर्ती प्रक्रियाएं ===

अर्ध-स्थैतिक प्रक्रिया | अर्ध-स्थैतिक और [[प्रतिवर्ती प्रक्रिया (थर्मोडायनामिक्स)|प्रतिवर्ती प्रक्रिया (ऊष्मागतिक्स)]] प्रक्रियाओं के लिए, [[ऊष्मप्रवैगिकी का पहला नियम]] है:

:<math>dU=\delta Q - \delta W</math>
जहाँ δ''Q'' तंत्र को आपूर्ति की गई ऊष्मा है और δ''W'' निकाय द्वारा किया गया कार्य है।

=== ऊष्मागतिक क्षमता ===
{{main|ऊष्मागतिक विभव}}
{{See also|Maxwell relations}}

निम्नलिखित ऊर्जाओं को [[थर्मोडायनामिक क्षमता|ऊष्मागतिक क्षमता]] कहा जाता है,

:{{table of thermodynamic potentials}}

और संबंधित मूलभूत ऊष्मागतिक संबंध या मास्टर समीकरण<ref>Physical chemistry, P.W. Atkins, Oxford University Press, 1978, {{ISBN|0 19 855148 7}}</ref> हैं:

{| class="wikitable"
|-
! Potential 
! Differential
|-
! Internal energy 
|<math>dU\left(S,V,{N_{i}}\right) = TdS - pdV + \sum_{i} \mu_{i} dN_i</math>
|-
! Enthalpy
|<math>dH\left(S,p,{N_{i}}\right) = TdS + Vdp + \sum_{i} \mu_{i} dN_{i}</math>
|-
!Helmholtz free energy
|<math>dF\left(T,V,{N_{i}}\right) = -SdT - pdV + \sum_{i} \mu_{i} dN_{i}</math>
|-
!Gibbs free energy
|<math>dG\left(T,p,{N_{i}}\right) = -SdT + Vdp + \sum_{i} \mu_{i} dN_{i}</math>
|-
|}


=== मैक्सवेल के संबंध ===

मैक्सवेल के चार सबसे आम संबंध हैं:

{| class="wikitable"
|-
! scope="col" width="100" | Physical situation
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equations
|-
!Thermodynamic potentials as functions of their natural variables
|{{plainlist}}
*<math>U(S,V)\,</math> = [[Internal energy]]
*<math>H(S,P)\,</math> = [[Enthalpy]]
*<math>F(T,V)\,</math> = [[Helmholtz free energy]]
*<math>G(T,P)\,</math> = [[Gibbs free energy]] 
{{endplainlist}}
|<math> \left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V = \frac{\partial^2 U }{\partial S \partial V} </math>

<math> \left(\frac{\partial T}{\partial P}\right)_S = +\left(\frac{\partial V}{\partial S}\right)_P = \frac{\partial^2 H }{\partial S \partial P}
</math>

<math> +\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V = - \frac{\partial^2 F }{\partial T \partial V} </math>

<math> -\left(\frac{\partial S}{\partial P}\right)_T = \left(\frac{\partial V}{\partial T}\right)_P = \frac{\partial^2 G }{\partial T \partial P} </math>
|-
|}
अधिक संबंधों में निम्नलिखित शामिल हैं।

{| class="wikitable"
|-
|<math> \left ( {\partial S\over \partial U} \right )_{V,N} = { 1\over T } </math>
|<math> \left ( {\partial S\over \partial V} \right )_{N,U} = { p\over T } </math>
|<math> \left ( {\partial S\over \partial N} \right )_{V,U} = - { \mu \over T } </math>
|-
|<math> \left ( {\partial T\over \partial S} \right )_V = { T \over C_V } </math>
|<math> \left ( {\partial T\over \partial S} \right )_P = { T \over C_P } </math>
|
|-
|<math> -\left ( {\partial p\over \partial V} \right )_T = { 1 \over {VK_T} } </math>
|
|
|-
|}
अन्य अंतर समीकरण हैं:

{| class="wikitable"
|-
! Name
! ''H''
! ''U''
! ''G''
|-
![[Gibbs–Helmholtz equation]]
|<math>H = -T^2\left(\frac{\partial \left(G/T\right)}{\partial T}\right)_p</math>
|<math>U = -T^2\left(\frac{\partial \left(F/T\right)}{\partial T}\right)_V</math>
|<math>G = -V^2\left(\frac{\partial \left(F/V\right)}{\partial V}\right)_T</math>
|-
|
|<math>\left(\frac{\partial H}{\partial p}\right)_T = V - T\left(\frac{\partial V}{\partial T}\right)_P</math>
|<math>\left(\frac{\partial U}{\partial V}\right)_T = T\left(\frac{\partial P}{\partial T}\right)_V - P</math>
|
|-
|}


=== क्वांटम गुण ===

*<math> U = N k_B T^2 \left(\frac{\partial \ln Z}{\partial T}\right)_V ~</math>
*<math> S =  \frac{U}{T} + N k_B \ln Z - N k \ln N + Nk ~</math> अप्रभेद्य कण

जहाँ N कणों की संख्या है, ''h'' प्लैंक नियतांक है, ''I'' जड़त्वाघूर्ण है, और ''Z'' विभिन्न रूपों में विभाजन फलन (सांख्यिकीय यांत्रिकी) है:

{| class="wikitable"
|-
!Degree of freedom
!Partition function
|-
!Translation
|<math> Z_t = \frac{(2 \pi m k_B T)^\frac{3}{2} V}{h^3} </math>
|-
!Vibration
|<math> Z_v = \frac{1}{1 - e^\frac{-h \omega}{2 \pi k_B T}} </math>
|-
!Rotation
|<math> Z_r = \frac{2 I k_B T}{\sigma (\frac{h}{2 \pi})^2} </math>

{{plainlist}}
*where:
*σ = 1 ([[heteronuclear molecule]]s)
*σ = 2 ([[homonuclear]])
{{endplainlist}}
|}


== पदार्थ के ऊष्मीय गुण ==

{| class="wikitable"
|-
! Coefficients 
! Equation
|-
![[Joule–Thomson effect|Joule-Thomson coefficient]]
|<math>\mu_{JT} = \left(\frac{\partial T}{\partial p}\right)_H</math>
|-
![[Compressibility]] (constant temperature)
|<math> K_T = -{ 1\over V } \left ( {\partial V\over \partial p} \right )_{T,N} </math>
|-
! [[Coefficient of thermal expansion]] (constant pressure)
|<math>\alpha_{p} = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_p</math>
|-
! Heat capacity (constant pressure)
| <math>C_p
= \left ( {\partial Q_{rev} \over \partial T} \right )_p
= \left ( {\partial U \over \partial T} \right )_p + p \left ( {\partial V \over \partial T} \right )_p 
= \left ( {\partial H \over \partial T} \right )_p
= T \left ( {\partial S \over \partial T} \right )_p </math>
|-
! Heat capacity (constant volume)
| <math>C_V
= \left ( {\partial Q_{rev} \over \partial T} \right )_V
= \left ( {\partial U \over \partial T} \right )_V
= T \left ( {\partial S \over \partial T} \right )_V </math>
|-
|}

{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Derivation of heat capacity (constant pressure)
|-
|
Since

:<math>
\left(\frac{\partial T}{\partial p}\right)_H
\left(\frac{\partial p}{\partial H}\right)_T
\left(\frac{\partial H}{\partial T}\right)_p
= -1
</math>

:<math>
\begin{align}
\left(\frac{\partial T}{\partial p}\right)_H 
& = -\left(\frac{\partial H}{\partial p}\right)_T
   \left(\frac{\partial T}{\partial H}\right)_p
\\
& = \frac{-1}{\left(\frac{\partial H}{\partial T}\right)_p}
  \left(\frac{\partial H}{\partial p}\right)_T
\end{align}
</math>

:<math>C_p = \left(\frac{\partial H}{\partial T}\right)_p</math>

:<math>
\Rightarrow \left(\frac{\partial T}{\partial p}\right)_H
= -\frac{1}{C_p}
   \left(\frac{\partial H}{\partial p}\right)_T
</math>
|}

{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Derivation of heat capacity (constant volume)
|-
|
Since

:<math>dU = \delta Q_{rev} - \delta W_{rev} ,</math>

(where δ''W''<sub>rev</sub> is the work done by the system),

:<math>\delta S = \frac{\delta Q_{rev}}{T}, \delta W_{rev}= p \delta V </math>

:<math> d U = T \delta S- p\delta V </math>

:<math>
\left(\frac{\partial U}{\partial T}\right)_V
= T\left(\frac{\partial S}{\partial T}\right)_V
- p\left(\frac{\partial V}{\partial T}\right)_V ; C_V = \left(\frac{\partial U}{\partial T}\right)_V
</math>

:<math>\Rightarrow C_V = T\left(\frac{\partial S}{\partial T}\right)_V</math>
|}


===तापीय स्थानांतरण===

{| class="wikitable"
|-
! scope="col" width="100" | Physical situation
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equations
|-
!Net intensity emission/absorption
| {{plainlist}}
*''T''<sub>external</sub> = external temperature (outside of system)
*''T''<sub>system</sub> = internal temperature (inside system)
*''ε'' = emmisivity
{{endplainlist}}
|<math> I = \sigma \epsilon \left ( T_\mathrm{external}^4 - T_\mathrm{system}^4 \right ) \,\!</math>
|-
!Internal energy of a substance
| {{plainlist}}
*''C<sub>V</sub>'' = isovolumetric heat capacity of substance
*Δ''T'' = temperature change of substance
{{endplainlist}}
|<math>\Delta U = N C_V \Delta T\,\!</math>
|-
!Meyer's equation
|{{plainlist}}
*''C<sub>p</sub>'' = isobaric heat capacity
*''C<sub>V</sub>'' = isovolumetric heat capacity
*''n'' = number of moles
{{endplainlist}}
|<math> C_p - C_V = nR \,\!</math>
|-
!Effective thermal conductivities
| {{plainlist}}
*''λ<sub>i</sub>'' = thermal conductivity of substance ''i''
*''λ''<sub>net</sub> = equivalent thermal conductivity  
{{endplainlist}}
| Series
<math> \lambda_\mathrm{net} = \sum_j \lambda_j \,\!</math>

Parallel
<math> \frac{1}{\lambda}_\mathrm{net} = \sum_j \left ( \frac{1}{\lambda}_j \right ) \,\!</math>
|-
|}


=== तापीय क्षमता ===

{| class="wikitable"
|-
! scope="col" width="100" | Physical situation
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equations
|-
!Thermodynamic engines
|{{plainlist}}
* ''η'' = efficiency
* ''W'' = work done by engine
* ''Q<sub>H</sub>'' = heat energy in higher temperature reservoir
* ''Q<sub>L</sub>'' = heat energy in lower temperature reservoir
* ''T<sub>H</sub>'' = temperature of higher temp. reservoir
* ''T<sub>L</sub>'' = temperature of lower temp. reservoir
{{endplainlist}}
|Thermodynamic engine:<br />
<math>\eta = \left |\frac{W}{Q_H} \right|\,\!</math>

Carnot engine efficiency:<br />
<math>\eta_c = 1 - \left | \frac{Q_L}{Q_H} \right | = 1-\frac{T_L}{T_H}\,\!</math>
|-
!Refrigeration
| ''K'' = coefficient of refrigeration performance
|Refrigeration performance
<math>K = \left | \frac{Q_L}{W} \right | \,\!</math>

Carnot refrigeration performance
<math>K_C = \frac{|Q_L|}{|Q_H|-|Q_L|} = \frac{T_L}{T_H-T_L}\,\!</math>
|-
|}

== यह भी देखें ==
{{colbegin}}
* [[एंटोनी समीकरण]]
* [[बेजान संख्या]]
* [[बोवेन अनुपात]]
*ब्रिजमैन के थर्मोडायनामिक समीकरण|ब्रिजमैन के समीकरण
* क्लॉसियस-क्लैपेरॉन संबंध
* [[प्रस्थान समारोह]]
* डुहेम-मार्ग्यूल्स समीकरण
* एहरेनफेस्ट समीकरण
* गिब्स-हेल्महोल्त्ज़ समीकरण
* [[चरण नियम]]
* कोप्प का नियम
*संबंधित राज्यों का नोरो-फ्रेनकेल नियम
*ऑनसेजर पारस्परिक संबंध
* [[ स्टीफन संख्या ]]
* [[ट्रिपल उत्पाद नियम]]
*[[सटीक अंतर]]{{colend}}

==संदर्भ==
<references/>
{{refbegin|2}}
* [[Peter Atkins|Atkins, Peter]] and de Paula, Julio  ''Physical Chemistry'', 7th edition, W.H. Freeman and Company, 2002 {{ISBN|0-7167-3539-3}}.
** Chapters 1–10, Part 1: "Equilibrium".
* {{cite journal | last=Bridgman | first=P. W. | title=A Complete Collection of Thermodynamic Formulas | journal=Physical Review | publisher=American Physical Society (APS) | volume=3 | issue=4 | date=1 March 1914 | issn=0031-899X | doi=10.1103/physrev.3.273|url=https://babel.hathitrust.org/cgi/pt?id=uc1.31210014450082&view=1up&seq=289 | pages=273–281}}
*Landsberg, Peter T.  <u>Thermodynamics and Statistical Mechanics</u>. New York: Dover Publications, Inc., 1990. ''(reprinted from Oxford University Press, 1978)''.
* Lewis, G.N., and Randall, M., "Thermodynamics", 2nd Edition, McGraw-Hill Book Company, New York, 1961.
* [[Linda Reichl|Reichl, L.E.]], ''A Modern Course in Statistical Physics'', 2nd edition, New York: John Wiley & Sons, 1998.
*Schroeder, Daniel V. ''Thermal Physics''.  San Francisco: Addison Wesley Longman, 2000 {{ISBN|0-201-38027-7}}.
*Silbey, Robert J., et al. ''Physical Chemistry'', 4th ed. New Jersey: Wiley, 2004.
*Callen, Herbert B. (1985). ''Thermodynamics and an Introduction to Themostatistics'', 2nd edition, New York: John Wiley & Sons.
{{refend}}


==बाहरी संबंध==
*[http://www.fxsolver.com/browse/?oc=3&cat=6&formulas=on Thermodynamic equation calculator]

{{Physics-footer}}
[[Category: थर्मोडायनामिक समीकरण]] 



[[Category: Machine Translated Page]]
[[Category:Created On 24/03/2023]]