Change in internal energy

Question

How do I calculate the change in internal energy in a closed system, where no heat is added? I need to know how to calculate the work done by the system. An example would be an ice cube in a closed jar.

Answer 1

A process where no heat is added is called adiabatic. It is a reversible process and one good example is to build a cylinder closed on top with a piston. Inside the cylinder we add some gas. We add an infinitesimal amount of mass and wait for the equilibrium to be estabilished. When this happens, we add another infinitesimal amount of mass and so on. This would be an adiabiatic compression.

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Calculating the work in an adiabatic process is not an easy task in the general case. You're not being very specific about what you want, so in my answer i'm going to prove the most common case which is the work done by an ideal gas in a adiabatic expansion, an then try to get to a conclusion that may help you.

Frist, we know that $\gamma$ is a constant that depends on the gas's nature. For example: For air you got $\gamma \approx 1.4$. We can easily verify that in an adiabatic process, the expression $PV^{\gamma}$ remains constant, where $P$ is the pressure and $V$ is the volume of the gas. Let $i$ and $f$ be indicatives of the initial and final state of the gas, respectively. We want to calculate: $$W = \int_{i}^{f} P \mathbb{d}V$$ And since $PV^{\gamma} = P_i V^{\gamma}_i = P_f V_f ^{\gamma} = K$, then the pressure $P$ with respect to volume at any part of the process is $P=\frac{K}{V^{\gamma}}$. Plugging in the integral: $$W = \int_{i}^{f} KV^{-\gamma}\mathbb{d}V = K \int_{i}^{f} V^{-\gamma}\mathbb{d}V = \frac{KV_f^{1-\gamma}}{1-\gamma} - \frac{KV_i^{1-\gamma}}{1-\gamma}$$ And since $K$ equals both $P_i V_i^{\gamma}$ and $P_f V_f^{\gamma}$: $$W = \frac{P_f V_f - P_i V_i}{1-\gamma}$$

And the most important thing here is that, unlike many other processes, the work here does not depend on the path. Therefore, one can calculate the work only knowing the final and the initial conditions of the system. There is no formula for a "general case" of adiabatic process. You must be more specific about what you want.