Q 57. - Calculus

Question

The ideal gas law states that the pressure, P , volume, V, temperature in degrees Kelvin, T, and amount, n, of a gas are related by the equation P = nRT/ V , where R is the universal gas constant. In Exercises 57–60, assume that the amount of the gas is constant and evaluate the specified limits. 

If the volume is held fixed, what is $\underset{T\to 0}{\lim }P$? Explain why this makes sense. 

Step-by-Step Solution

Answer

The value of the $\underset{T\to 0}{\lim }P$ is 0.

This makes sense because when the gas cools, the contact between the gas particles reduces fast, lowering the pressure.

Step 1. Given information.

We have given expression: $\underset{T\to 0}{\lim }P$.

Step 2: Evaluate the specified limits. Explain why this makes sense.

$$\begin{gathered} \underset{T\to 0}{\lim }P=\underset{T\to 0}{\lim }\frac{nRT}{V} \\ =\frac{nR}{V}\underset{T\to 0}{\lim }T \\ =0 \end{gathered}$$

Therefore, the value of $\underset{T\to 0}{\lim }P$ is 0 when$V$ is constant.

This predicts that as the temperature of a gas is reduced to nearly zero at constant volume, the pressure P of the gas will similarly approach zero.

This makes sense as the gas is cooling off and hence the interaction between the gas particles decreases rapidly, causing the pressure also to drop.