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4.E: Putting the First Law to Work (Exercises)

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    Given the relationship

    \[ \left( - \dfrac{\partial U}{\partial V} \right)_T= T\left( - \dfrac{\partial p}{\partial T} \right)_V-p \]

    show that

    \[\left( - \dfrac{\partial U}{\partial V} \right)_T =0 \]

    for an ideal gas.


    Determine if the following differential is exact, and if so, find the function \(z(x, y)\) that satisfies the expression.

    \[ dz = 4xy\,dz + 2x^2 dy\]


    For a van der Waals gas,

    \[\left(\dfrac{\partial U}{\partial V}\right)_T = \left(\dfrac{an^2}{V^2}\right) \]

    Find an expression in terms of \(a\), \(n\), \(V\), and \(R\) for

    \[\left(\dfrac{\partial T}{\partial V}\right)_U\]

    if \(C_V = 3/2 R\). Use the expression to calculate the temperature change for 1.00 mol of Xe (a = 4.19 atm L2 mol -2) expanding adiabatically against a vacuum from 10.0 L to 20.0 L.


    Given the following data, calculate the change in volume for 50.0 cm3 of

    1. neon and
    2. copper

    due to an increase in pressure from 1.00 atm to 0.750 atm at 298 K.

    Substance T (at 1.00 atm and 298 K)
    Ne 1.00 atm-1
    Cu 0.735 x 10-6 atm-1


    Consider a gas that follows the equation of state

    \[ p =\dfrac{nRT}{V-nb}\]

    derive an expression for

    1. the isobaric thermal expansivity, \(\alpha\)
    2. the Joule-Thomson coefficient, \(\mu_{JT}\)

    \[\mu_{JT} = \dfrac{V}{C_p} (T \alpha -1)\]



    \[ \left(\dfrac{\partial H}{\partial p}\right)_T = -T \left(\dfrac{\partial V}{\partial T}\right)_p +V \]

    derive an expression for

    \[\left(\dfrac{\partial U}{\partial p}\right)_T \]

    in terms of measurable properties. Use your result to calculate the change in the internal energy of 18.0 g of water when the pressure is increased from 1.00 atm to 20.0 atm at 298 K.


    Derive an expression for

    \[\left(\dfrac{\partial U}{\partial T}\right)_p \]

    Begin with the definition of enthalpy, in order to determine

    \[ dH = dU + pdV + Vdp\]

    Finish by dividing by dT and constraining to constant pressure. Make substitutions for the measurable quantities, and solve for

    \[\left(\dfrac{\partial U}{\partial T}\right)_p .\]


    Derive an expression for the difference between \(C_p\) and \(C_V\) in terms of the internal pressure, \(\alpha\), \(p\) and \(V\). Using the definition for \(H\) as a starting point, show that

    \[\left(\dfrac{\partial H}{\partial T}\right)_p = \left(\dfrac{\partial U}{\partial T}\right)_p + p \left(\dfrac{\partial V}{\partial T}\right)_p \]

    Now, find an expression for by starting with \(U(V,T)\) and writing an expression for the total differential \(dU(V,T)\).

    Divide this expression by \(dp\) and constrain to constant \(T\). Substitute this into the previous expressions and solve for

    \[\left(\dfrac{\partial G}{\partial T}\right)_p - \left(\dfrac{\partial U}{\partial T}\right)_V .\]


    Evaluate the expression you derived in problem 8 for an ideal, assuming that the internal pressure of an ideal gas is zero.

    Contributors and Attributions

    • Patrick E. Fleming (Department of Chemistry and Biochemistry; California State University, East Bay)

    4.E: Putting the First Law to Work (Exercises) is shared under a not declared license and was authored, remixed, and/or curated by Patrick Fleming.

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