A certain small quantity is given by CV d T+(R T)/(V) d V where CV and R are constants and T a

step 1 So what we have here is the following DQ is CVDT, add RT, D, LNV. So that's just to offer you so

A certain small quantity is given by CV d T+(R T)/(V) d V...

A certain small quantity is given by $$ C_V \mathrm{~d} T+\frac{R T}{V} \mathrm{~d} V $$ where $C_V$ and $R$ are constants and $T$ and $V$ are functions of state. (i) Show that this small quantity is an improper differential. (ii) Let $\mathrm{d} Q=C_V \mathrm{~d} T+$ $\frac{R T}{V} \mathrm{~d} V$. Show that $\mathrm{d} Q / T$ is a proper differential.

A certain small quantity is given by
$$
C_V \mathrm{~d} T+\frac{R T}{V} \mathrm{~d} V
$$
where $C_V$ and $R$ are constants and $T$ and $V$ are functions of state. (i) Show that this small quantity is an improper differential. (ii) Let $\mathrm{d} Q=C_V \mathrm{~d} T+$ $\frac{R T}{V} \mathrm{~d} V$. Show that $\mathrm{d} Q / T$ is a proper differential.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Integrating Factor

An integrating factor is a function that, when multiplied by an inexact differential, converts it into an exact differential. In thermodynamics, the heat differential ?Q is generally inexact, but when divided by temperature T (i.e., ?Q/T), it becomes exact, representing the differential of entropy. This procedure helps in identifying and working with state functions and is a useful method in thermodynamic derivations.

Inexact Differential

An inexact differential does not correspond to the total differential of any state function. Its integral depends on the path between the initial and final states, which is typical for quantities like heat and work in thermodynamics. Recognizing an inexact differential requires checking if the conditions for exactness (equality of cross partial derivatives) fail, indicating that the property does not have state function status.

Exact Differential

An exact differential is a differential expression that corresponds to the total differential of some state function. In other words, there exists a function U(T, V) such that dU = M(T, V)dT + N(T, V)dV, and the mixed partials satisfy ?M/?V = ?N/?T. This guarantees that the integration of the differential from one state to another is independent of the path taken, a hallmark of state functions in thermodynamics.

Transcript

Please give Ace some feedback