The Isochronicles: This First Law Saga Is a Gas

DOFPro Team

Introduction

1st Law behavior for Ideal Gases for:

  • Isothermal Process
  • Isobaric Process
  • Isochoric Process
  • Isentropic Process

Equations and concepts for dealing with ideal gases in first law problems

First: heat capacity doesn’t vary with pressure for ideal gases, only with temperature.

\[d\hat{U} = C_v dT,\hspace{120px} \Delta \hat{U} = \int\limits_{T_1}^{T_2} C_v dT\]

\[d\hat{H} = C_p dT,\hspace{120px} \Delta \hat{H} = \int\limits_{T_1}^{T_2} C_p dT\]

\[C_p = C_v + R\]

Second: for a closed system with ideal gas and no change in kinetic or potential energy.

\[d\hat{U} = \frac{dQ}{n} + \frac{dW}{n}\]

\[C_v dT = \frac{dQ}{n} - Pd\hat{V}\]

\[dQ = nC_v dT + PdV = nC_v dT + nRT \frac{dV}{V}\]

\[dQ = nC_p dT - nRT \frac{dP}{P}\]

Isothermal (Constant Temperature) Process:

\[d\hat{U} = d\hat{H} = 0\]

\[dQ = dQ\]

\[dQ = n C_v dT + nRT \frac{dV}{V} = n C_p dT - nRT \frac{dP}{P}\]

\[Q = -W = nRT \ln \frac{V_2}{V_1} = nRT \ln \frac{P_1}{P_2}\]

Note the order of the subscripts on \(V\) and \(P\).

Isobaric (Constant Pressure) Process:

\(d\hat{U} = C_v dT\)

\[\Delta \hat{U} = \int\limits_{T_1}^{T_2}C_v dT\]

\(dW = -PdV\)

\[\begin{align} W &= -P(V_2 - V_1) \\&= -nR(T_2 - T_1) \end{align}\]

\(\Delta U = Q + W = Q - P \Delta V\)

\(\Delta U + P \Delta V = Q\)

\[Q = \Delta H = n \int\limits_{T_1}^{T_2}C_p dT\]

Isochoric (Constant Volume) Process:

\[dW = -PdV\]

\[d \hat{U} = C_V dT\]

\[Q = \Delta U = n \int\limits_{T_1}^{T_2}C_v dT\]

Adiabatic Reversible (Isentropic) Process:

\[dQ = n C_v dT + P dV = n C_V dT + nRT \frac{dV}{V}\]

\[n C_v dT = -nRT \frac{dV}{V}\]

\[\frac{dT}{T} = - \frac{R}{C_v} \frac{dV}{V}\]

If \(C_v = \mathrm{constant}\)

\[\ln \frac{T_2}{T_1} = \frac{R}{C_v} \ln {\frac{V_1}{V_2}}\]

\[\frac{T_2}{T_1} = \left(\frac{V_1}{V_2}\right)^{\frac{R}{C_v}}\]

\[\frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{\frac{R}{C_p}}\]

Note the order of the subscripts on \(V\) and \(P\).

The Takeaways

  1. The Internal Energy (\(U\)), Enthalpy (\(H\)), and Heat Capacity (\(C_v,\ C_p\)) of an ideal gas are functions of temperature only, not pressure.
  2. We developed the relationship among \(Q\), \(W\), \(T\), \((V_2/V_1)\), and \((P_1/P_2)\) for an isothermal process for an ideal gas.
  3. We developed the relationship among \(Q\), \(W\), \(\Delta U\), \(\Delta H\), \(T\), \(V\), and \(P\) for an isobaric process for an ideal gas.
  4. We developed the relationship between \(Q\) and \(\Delta U\) for an isochoric process for an ideal gas.
  5. We developed the relationship among \(W\), \((T_2/T_1)\), \((V_1/V_2)\), and \((P_2/P_1)\) for an isentropic process for an ideal gas.






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