Question

According to the Ideal Gas Law, \(P V=k T,\) where \(P\) is pressure, \(V\) is volume, \(T\) is temperature (in Kelvins), and \(k\) is a constant of proportionality. A tank contains 2600 cubic inches of nitrogen at a pressure of 20 pounds per square inch and a temperature of \(300 \mathrm{~K}\). (a) Determine \(k\). (b) Write \(P\) as a function of \(V\) and \(T\) and describe the level curves.

Short answer

The constant of proportionality k can be calculated using the given values and the formula \(k = \frac{P V}{T}\). From this, we can reformulate the ideal gas law as \(P = \frac{k T}{V}\), setting P as a function of V and T. The level curves for this function in the PV plane are hyperbolas, as P varies inversely with V for a fixed value of T.

Step-by-step solution

Calculating the constant of proportionality

First, use the values of P (pressure), V (volume), and T (temperature) given in the exercise to calculate the constant of proportionality, k. We know that:\(P V=k T\)So, rearranging for k we have:\(k = \frac{P V}{T}\)Substitute P = 20 pounds per square inch, V = 2600 cubic inches, and T = 300 K into the formula to find k.

Deriving the equation for P in terms of V and T

Now, we have the constant of proportionality k, and we can rewrite the ideal gas law with P as a function of V and T. We just rearrange the Ideal Gas Law to solve for P:\(P = \frac{k T}{V}\)Substitute the obtained value of k in to this formula. Now, we have the equation for P in terms of V and T.

Describing the level curves

The level curves of this function can be described by fixing T (temperature), and examining the function \(P = \frac{k T}{V}\) as a function of V. The level curves in the PV plane are hyperbolas with P and V as the axes, because for any fixed value of T, P varies inversely with V.