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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
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    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)
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    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
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    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
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    }\\\]

    \[\ {}_{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
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    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
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    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
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    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
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    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}=

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    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}}}=
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    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
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    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
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    }\\ \]

    (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
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    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
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    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
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    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)

    This page titled 4.7: Key equations is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Claire Yu Yan (BC Campus) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.