Problem 20 - University Physics with Modern Physics

Question

A relationship that gives the pressure, \(p\), of a substance as a function of its density, \(\rho\), and temperature, \(T\), is called an equation of state. For a gas with molar mass \(M\), write the Ideal Gas Law as an equation of state.

Step-by-Step Solution

Answer

Answer: The equation of state of an ideal gas in terms of density (ρ), temperature (T), and molar mass (M) is: p = ρRT/M

Step 1: Write down the Ideal Gas Law

The Ideal Gas Law is given by: PV = nRT where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature.

Step 2: Substitute the number of moles using mass and molar mass

We know that n (number of moles) can be written as the mass (m) divided by the molar mass (M). Substituting m/M for n, we get: PV = (m/M)RT

Step 3: Rewrite mass in terms of density and volume

Since density (ρ) is defined as mass (m) divided by volume (V), we can represent mass as the product of density and volume, that is m = ρV. Substituting ρV for m, we get: PV = (ρV/M)RT

Step 4: Simplify the equation

Notice that the volume (V) appears on both sides of the equation. We can divide both sides by volume to isolate the pressure (P): P = ρRT/M

Step 5: Final equation of state

The Ideal Gas Law in terms of density (ρ), pressure (p), temperature (T), and molar mass (M) is: p = ρRT/M