Understanding the Ideal Gas Law Partial Derivative

What is the partial derivative of pressure (P) with respect to volume (V) at constant temperature (T) in the ideal gas law PV = RT?

Partial Derivative Calculation

The partial derivative of P with respect to V at constant T in the ideal gas law PV = RT, is -RT/V².

The given equation represents the Ideal Gas Law, denoted as PV = RT, where 'P' stands for pressure, 'V' for volume, 'R' is the universal gas constant, and 'T' is the temperature in Kelvin. In this formula, we assume 'n' number of moles is equal to 1. In order to find the partial derivative of P with respect to V at constant T for an ideal gas, you need to rearrange the formula to P = RT/V and differentiate it with respect to 'V'.

Upon differentiating, we will get -RT/V². Therefore, the partial derivative of P with respect to V at constant T for an ideal gas is -RT/V². This implies that the pressure of an ideal gas decreases with an increase in volume, holding the temperature constant. This concept aligns with Boyle's Law, stating that pressure is inversely proportional to volume for a fixed mass of gas at a constant temperature.

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