4.7: Key equations

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Key equations

Constant-volume specific heat	$C_v=\left(\displaystyle\frac{\partial u}{\partial T}\right)_v$
Change in specific internal energy for all fluids	$\Delta u = u_2-u_1$
Change in specific internal energy for ideal gases	$\Delta u = C_v\left(T_2-T_1\right)$
Specific heat transfer	$q=\displaystyle\frac{Q}{m}$
Boundary work	${}_{1}W_{2}=\displaystyle\int_{1}^{2}{Pd\mathbb{V}\ }$
Specific boundary work	${}_{1}w_{2}=\displaystyle\int_{1}^{2}{Pdv\ }$
Spring force	$F=Kx$
Spring work	$W_{spring}=\displaystyle\int_{1}^{2}{Fdx=}\displaystyle\frac{1}{2}K\left(x_2^2-x_1^2\right)$
The first law of thermodynamics for closed systems	$\Delta U = U_2-U_1 = {}_{1}Q_{2} - {}_{1}W_{2}$ , assuming $\Delta KE = \Delta PE = 0$

Equations for polytropic Processes

Process function	${P}{v}^{n}= \rm{constant}$
Boundary work for real gases	If $n \neq 1$ , \[{}_{1}W_{2}=\displaystyle\frac ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[2]/td/p[1]/span, line 1, column 2 {1-n}\\\] If $\ n=1,$ \[{}_{1}W_{2}={P}_\mathbf{1}\mathbb{V}_\mathbf{1}{ln}{\displaystyle\frac{\mathbb{V}_\mathbf{2}}{\mathbb{V}_\mathbf{1}}}={P}_\mathbf{2}\mathbb{V}_\mathbf{2}{ln}{\displaystyle\frac{\mathbb{V}_\mathbf{2}}{\mathbb{V}_\mathbf{1}}}\\ \] \[{}_{1}W_{2}={P}_\mathbf{1}\mathbb{V}_\mathbf{1}{ln}{\displaystyle\frac ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[2]/td/p[4]/span[1], line 1, column 2 ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[2]/td/p[4]/span[2], line 1, column 2 }={P}_\mathbf{2}\mathbb{V}_\mathbf{2}{ln}{\displaystyle\frac ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[2]/td/p[4]/span[3], line 1, column 2 ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[2]/td/p[4]/span[4], line 1, column 2 }\\ \]
Specific boundary work for real gases	If $n \neq 1$ \[\ {}_{1}w_{2}=\displaystyle\frac ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[3]/td/p[1]/span, line 1, column 2 {1-n}\\\] If $\ n=1,$ \[\ {}_{1}w_{2}={P}_\mathbf{1}{v}_\mathbf{1}{ln}{\displaystyle\frac ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[3]/td/p[3]/span[1], line 1, column 2 ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[3]/td/p[3]/span[2], line 1, column 2 }={P}_\mathbf{2}{v}_\mathbf{2}{ln}{\displaystyle\frac ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[3]/td/p[3]/span[3], line 1, column 2 ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[3]/td/p[3]/span[4], line 1, column 2 }\\\] \[\ {}_{1}w_{2}={P}_\mathbf{1}{v}_\mathbf{1}{ln}{\displaystyle\frac ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[3]/td/p[4]/span[1], line 1, column 2 ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[3]/td/p[4]/span[2], line 1, column 2 }={P}_\mathbf{2}{v}_\mathbf{2}{ln}{\displaystyle\frac ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[3]/td/p[4]/span[3], line 1, column 2 ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[3]/td/p[4]/span[4], line 1, column 2 }\\\]
Boundary work for ideal gases	If $n \neq 1$ \[{}_{1}W_{2}=\displaystyle\frac ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[4]/td/p[1]/span, line 1, column 2 {1-n}\\\] If $\ n=1,$ \[{}_{1}W_{2}={P}_\mathbf{1}\mathbb{V}_\mathbf{1}{ln}{\displaystyle\frac{\mathbb{V}_\mathbf{2}}{\mathbb{V}_\mathbf{1}}}={P}_\mathbf{2}\mathbb{V}_\mathbf{2}{ln}{\displaystyle\frac{\mathbb{V}_\mathbf{2}}{\mathbb{V}_\mathbf{1}}}\\ \] \[{}_{1}W_{2}={P}_\mathbf{1}\mathbb{V}_\mathbf{1}{ln}{\displaystyle\frac ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[4]/td/p[4]/span[1], line 1, column 2 ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[4]/td/p[4]/span[2], line 1, column 2 }={P}_\mathbf{2}\mathbb{V}_\mathbf{2}{ln}{\displaystyle\frac ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[4]/td/p[4]/span[3], line 1, column 2 ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[4]/td/p[4]/span[4], line 1, column 2 }\\ \] \[\ {}_{1}W_{2}= UndefinedNameError: reference to undefined name 'mRT' (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[4]/td/p[5]/span[1], line 1, column 1 {ln}{\displaystyle\frac{\mathbb{V}_\mathbf{2}}{\mathbb{V}_\mathbf{1}}}= UndefinedNameError: reference to undefined name 'mRT' (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[4]/td/p[5]/span[2], line 1, column 1 {ln}{\displaystyle\frac ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[4]/td/p[5]/span[3], line 1, column 2 ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[4]/td/p[5]/span[4], line 1, column 2 }\\ \] ( $T$ in Kelvin)
Specific boundary work for ideal gases	If $n \neq 1$ \[\ {}_{1}w_{2}=\displaystyle\frac ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[5]/td/p[1]/span, line 1, column 2 {1-n}\\\] If $\ n=1,$ \[\ {}_{1}w_{2}={P}_\mathbf{1}{v}_\mathbf{1}{ln}{\displaystyle\frac ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[5]/td/p[3]/span[1], line 1, column 2 ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[5]/td/p[3]/span[2], line 1, column 2 }={P}_\mathbf{2}{v}_\mathbf{2}{ln}{\displaystyle\frac ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[5]/td/p[3]/span[3], line 1, column 2 ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[5]/td/p[3]/span[4], line 1, column 2 }\\\] \[\ {}_{1}w_{2}={P}_\mathbf{1}{v}_\mathbf{1}{ln}{\displaystyle\frac ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[5]/td/p[4]/span[1], line 1, column 2 ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[5]/td/p[4]/span[2], line 1, column 2 }={P}_\mathbf{2}{v}_\mathbf{2}{ln}{\displaystyle\frac ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[5]/td/p[4]/span[3], line 1, column 2 ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[5]/td/p[4]/span[4], line 1, column 2 }\\\] \[\ {}_{1}w_{2}= UndefinedNameError: reference to undefined name 'RT' (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[5]/td/p[5]/span[1], line 1, column 1 {ln}{\displaystyle\frac ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[5]/td/p[5]/span[2], line 1, column 2 ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[5]/td/p[5]/span[3], line 1, column 2 }= UndefinedNameError: reference to undefined name 'RT' (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[5]/td/p[5]/span[4], line 1, column 1 {ln}{\displaystyle\frac ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[5]/td/p[5]/span[5], line 1, column 2 ParseError: EOF expected (click for details) Callstack: at (Bookshelves/Mechanical_Engineering/Introduction_to_Engineering_Thermodynamics_(Yan)/04:_The_First_Law_of_Thermodynamics_for_Closed_Systems/4.07:_Key_equations), /content/body/div[2]/table/tbody/tr[5]/td/p[5]/span[6], line 1, column 2 }\\\] ( $T$ in Kelvin)

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