# Changes of Internal Energy un open system

My professor says the internal energy of a system can be calculated as $U=n T c_v$. Now I always thought that we could only calculate a change in internal energy $\Delta U = n c_v \Delta T$.

Following this idea, he says that a recipient that contains gas and loses a certain number of moles at constant temperautre undergoes a change in internal energy of $\Delta U = T c_v \Delta n$.

I've never seen any of this in any book. I believe he is assuming that the internal energy at 0 Kelvin is zero, which I believe is not true. However, in the case of the recipient losing gas,I wouldn't know how to calculate $\Delta U$.

Any thoughts? thanks!

Suppose you assign an reference state for internal energy such that the internal energy per mole u of the gas is zero at $T=T_r$. This reference state is going to be held fixed for the material both entering or leaving the open system, as well as the material within the system at any time. Then, in the initial state of a constant-temperature system, $U_{init}=n_{init}C_v(T-T_r)$ and, in the final state, $U_{final}=n_{final}C_v(T-T_r)$. So, $$\Delta U=\Delta nC_v(T-T_r)$$Typically, in the overall energy balance on the system, including entering and exiting streams, $T_r$ will cancel out (just as it would for a closed system).
• Neglecting kinetic and potential energy effects, $$\frac{dU}{dt}=\dot{Q}-\dot{W}_s+\dot{m}_{in}h_{in}-\dot{m}_{out}h_{out}$$where $\dot{W}_s$ is the rate of doing shaft work. I'm sure you've seen this in some form before. – Chet Miller Feb 27 '16 at 15:03
• The value of $h{out}$ depends on what is being done. If the tank is adiabatic, Q dot is zero, and, if the gas exits through a valve, the value of $h{out}$ is the same as h in the tank (no enthalpy change across the valve) at the moment that the differential mass is leaving. If the tank is somehow held isothermal, Q dot is not zero, but u and h in the tank don't change, and $h{out}$ is equal to the (constant) h in the tank throughout the change. – Chet Miller Feb 28 '16 at 16:09