Question

The gas law for a fixed mass m of an ideal gas at absolute temperature T, pressure P, and volume V is PV = mRT, where R is the gas constant. Find the partial derivatives ∂P/∂V, ∂V/∂T and ∂T/∂P.

Answer 1

The partial derivatives of the ideal gas law equation PV = mRT with respect to volume, temperature, and pressure are ∂P/∂V = -mRT/V^2, ∂V/∂T = mR/P, and ∂T/∂P = V/(mR), respectively.

Step-by-step explanation:

The question asks for the partial derivatives ∂P/∂V, ∂V/∂T, and ∂T/∂P of the ideal gas law, which is expressed as PV = mRT. To find these derivatives, we need to apply the rules of calculus to the gas law equation, treating all other variables as constants when taking the derivative with respect to one.

To find ∂P/∂V, we will differentiate the ideal gas law with respect to V, assuming m, R, and T as constants. This yields:

∂P/∂V = -mRT/V2

For ∂V/∂T, we will differentiate the ideal gas law with respect to T, assuming m, R, and P as constants. The result is:

∂V/∂T = mR/P

Lastly, ∂T/∂P is found by differentiating with respect to P, holding m, R, and V constant, which leads to:

∂T/∂P = V/(mR)