Q 59. - 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$\underset{v\to 0}{\lim }P$ ? Explain why this makes sense 

Step-by-Step Solution

Answer

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

This makes sense as the gas is extremely compressed, the interaction between the gas particles increases rapidly, causing the pressure to rise infinitely without any bound

Step 1. Given information.

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

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

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

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

This predicts that when a gas is compressed to 0 volume and kept at constant temperature, the pressure P of the gas approaches infinity.

This makes sense because, when the gas is very compressed, the interaction between the gas particles intensifies rapidly, causing the pressure to climb inexorably.