Skip main navigation

Internal energy and enthalpy of ideal gases depend only on temperature

H(T) and U(T) of ideal gases
0.4
Previously, we said that the enthalpy of an ideal gas is independent of pressure at constant temperature. And the internal energy of an ideal gas is independent of volume at constant temperature. The enthalpy and internal energy of an ideal gas were asserted to be functions of temperature only. Here, we can prove it using the property relations. Let’s begin with enthalpy. dH is TdS + VdP. Divide both side by dP at constant temperature. Then (dH over dP) is T times (dS over dP) + V. So, to get the pressure dependence of enthalpy at constant temperature, we need (dS over dP) at constant temperature. Among the property relations, it can be obtained from dG, since G has variables of P and T.
63
(dS over dP) at constant T is - (dV over dT) at constant P. So, (dH over dP) is now - T (dV over dT) + V. For ideal gas, V equals to RT over P from the equation of state. Then, (dH over dP) at constant T becomes zero. Thus, enthalpy does not depend on pressure at constant T and it is a function of temperature only. Similarly, let’s prove that the internal energy of an ideal gas is a function of temperature only and independent of volume. dU is TdS - PdV. Divide both sides with dV at constant T. Then the volume dependence of the internal energy can be calculated from (dS over dV) at constant T.
131.3
This can be obtained from dF since F has T, V as variables. dF is - SdT - PdV. The property relation from here is (dS over dV) at constant T equals (dP over dT) at constant V. So, (dU over dV) is T times (dP over dV) - P. Again P is RT over V for the ideal gas. Thus (dU over dV) at constant T is zero. So the internal energy is a function of temperature only.

The internal energy and enthalpy of ideal gases depends only on temperature, not on volume or pressure.

We can prove these property of ideal gases using property relations. From the fundamental equations for internal energy and enthalpy, the volume dependence of internal energy and the pressure dependence of enthalpy for ideal gases are derived. By applying property relations, it is proved that the internal energy and enthalpy of ideal gases do not depend on volume and pressure, repectively.

This article is from the free online

Thermodynamics in Energy Engineering

Created by
FutureLearn - Learning For Life

Our purpose is to transform access to education.

We offer a diverse selection of courses from leading universities and cultural institutions from around the world. These are delivered one step at a time, and are accessible on mobile, tablet and desktop, so you can fit learning around your life.

We believe learning should be an enjoyable, social experience, so our courses offer the opportunity to discuss what you’re learning with others as you go, helping you make fresh discoveries and form new ideas.
You can unlock new opportunities with unlimited access to hundreds of online short courses for a year by subscribing to our Unlimited package. Build your knowledge with top universities and organisations.

Learn more about how FutureLearn is transforming access to education