Q 58. - 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$\underset{T\to \infty }{\lim }P$ ? Explain why this makes sense. 

Step-by-Step Solution

Answer

The value of the  $\underset{T\to \infty }{\lim }P$ is $\infty$. When $V$ is constant.

This makes sense because as the gas heats up, the interaction between the gas particles intensifies, causing the pressure to rise as well.

Step 1. Given information.

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

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

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

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

This implies that as the temperature of a gas rises without bound at constant volume, the pressure P of the gas rises to near infinity.

This makes sense because as the gas heats up, the interaction between the gas particles intensifies, causing the pressure to rise as well.