Q 60. - 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 temperature is held fixed, what is $\underset{v\to \infty }{\lim }P$? Explain why this makes sense 

Step-by-Step Solution

Answer

The value of $\underset{v\to \infty }{\lim }P$ is $0$.When $T$ is constant.

This makes sense because as the gas expands, the contact between the gas particles decreases rapidly, resulting in a rapid drop in pressure.

Step 1. Given information.

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

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

$$\begin{gathered} \underset{V\to \infty }{\lim }P=\underset{V\to \infty }{\lim }\frac{nRT}{V} \\ =nRT\underset{v\to \infty }{\lim }\frac{1}{V} \\ =0 \end{gathered}$$

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

This predicts that if a gas is allowed to expand endlessly with constant temperature, its pressure P will approach zero.

This makes sense because as the gas expands, the contact between the gas particles decreases rapidly, resulting in a rapid drop in pressure.