The problem states that according to the ideal gas law, the pressure $P$, temperature $T$, and volume $V$ of a gas are related by $PV = kT$, where $k$ is a constant. We are asked to find the rate of change of pressure with respect to temperature ($\frac{dP}{dT}$) when the temperature is $T = 300$ K and the volume is kept fixed at $V = 100$ cubic inches. The pressure is measured in pounds per square inch.

1. Problem Description

The problem states that according to the ideal gas law, the pressure $P$, temperature $T$, and volume $V$ of a gas are related by $PV = kT$, where $k$ is a constant. We are asked to find the rate of change of pressure with respect to temperature ($\frac{dP}{dT}$) when the temperature is $T = 300$ K and the volume is kept fixed at $V = 100$ cubic inches. The pressure is measured in pounds per square inch.

2. Solution Steps

The ideal gas law is given by:

$PV = kT$

We are asked to find the rate of change of pressure with respect to temperature, so we need to find $\frac{dP}{dT}$. Since the volume $V$ is constant and $k$ is a constant, we can differentiate both sides of the equation with respect to $T$:

$\frac{d}{dT}(PV) = \frac{d}{dT}(kT)$

$V \frac{dP}{dT} = k$

Now we can solve for $\frac{dP}{dT}$:

$\frac{dP}{dT} = \frac{k}{V}$

We are given that $V = 100$ cubic inches. Therefore,

$\frac{dP}{dT} = \frac{k}{100}$

Since the temperature does not appear in the derivative expression and only $V=100$, we simply need $\frac{k}{100}$ when $T = 300K$. Therefore,

$\frac{dP}{dT} = \frac{k}{100}$

3. Final Answer

The rate of change of pressure with respect to temperature is $\frac{k}{100}$.