Question

Using the ideal gas law, show that for an ideal gas of density \(\rho, \mathrm{P} / \rho \mathrm{T}\) is a constant.

Short answer

Using the ideal gas law \(PV = nRT\), we can express the number of moles \(n\) in terms of density \(\rho\), as \(n = \frac{\rho V}{M}\). Substituting this expression for n into the ideal gas law equation, we have \(PV = \left( \frac{\rho V}{M} \right) RT\). Now isolating the target ratio \(\frac{P}{\rho T}\), we get \(\frac{P}{\rho T} = \frac{MRT}{\rho V}\). Since the ideal gas constant (\(R\)) and molar mass (\(M\)) are constants, the ratio \(\frac{P}{\rho T}\) is indeed a constant for an ideal gas.

Step-by-step solution

Write down the ideal gas law equation

The ideal gas law equation is given by: \[PV = nRT\] where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.

Express the number of moles through density

The density of an ideal gas, denoted as ρ, is given by the ratio of the mass (m) to the volume (V): \[\rho = \frac{m}{V}\] Now, we know that the number of moles, n is the ratio of the mass (m) to the molar mass, M: \[n = \frac{m}{M}\] Combining these two expressions, we can write the number of moles as: \[n = \frac{\rho V}{M}\]

Substitute n in the ideal gas law equation

We will now substitute the expression for n we derived in step 2 into the ideal gas law equation: \[PV = \left( \frac{\rho V}{M} \right) RT\]

Simplify the equation and isolate the target ratio

We want to show that the ratio P/(ρT) is a constant. Let's isolate this ratio from the equation: \[\frac{P}{\rho T} = \frac{VMRT}{\rho V^2}\] Now cancel out the V in the numerator and denominator: \[\frac{P}{\rho T} = \frac{MRT}{\rho V}\]

Identify the constant

The ratio on the left side of the equation is a constant since R (the ideal gas constant) and M (molar mass) are constants. So, we can write the expression as: \[\frac{P}{\rho T} = \frac{MR}{\rho}\] Thus, for an ideal gas, the ratio of pressure to the product of density and temperature (P/ρT) is indeed a constant.

Key concepts

Understanding Gas Density in Relation to the Ideal Gas Law

When you hear about gas density, you're essentially diving into the measure of how much mass of a gas is contained in a given volume. For students grappling with the ideal gas law, understanding gas density is a fundamental step. It falls in the realm of the equation \rho = \frac{m}{V}\, where \(\rho\) represents the gas density, \(m\) indicates the mass, and \(V\) denotes the volume.

Visualize gas density as how tightly packed the gas molecules are within a certain space. It's like comparing a crowded room to an almost empty one - the more people (molecules) there are in the room (volume), the higher the density. In the context of the ideal gas law, we link density to the number of moles and the molar mass. When you rearrange the ideal gas equation to introduce density, you get a clear picture of how pressure, volume, and temperature interrelate with the amount of substance you’re dealing with.

Molar Mass: A Key Player in the Ideal Gas Equation

Onto one of the significant ingredients in the ideal gas equation: molar mass, denoted by \(M\). Molar mass acts like a bridge connecting the macroscopic properties of a gas we can measure (like density) to the microscopic world of atoms and molecules, represented by the mole concept.

To demystify molar mass, consider it the weight of one mole of a substance. It's usually expressed in grams per mole (g/mol) and can be thought of as the atomic or molecular 'price tag' telling you how many grams one mole of that substance weighs. A mole, just so you remember, is that famously large number, Avogadro’s number, amounting to approximately \(6.022 \times 10^{23}\) entities. Why does this matter in our exercise? Because knowing the molar mass allows us to convert between the mass of a gas and the number of moles, which is what we did when we derived \(n = \frac{\rho V}{M}\) in the step-by-step solution. It's like converting currency when traveling - if you're moving from mass to moles, you're going to need the molar mass exchange rate.

The Role of the Ideal Gas Constant in Gaseous Behaviors

Imagine a universal constant that serves as the heartbeat of the ideal gas law - that is the ideal gas constant, \(R\). It’s the proportionality factor that ties together pressure, volume, temperature, and the number of moles into one neat equation, \(PV = nRT\).

The value of \(R\) doesn’t change; it is a fixed number that encapsulates the physical properties of gases. Its consistent value across different gases and conditions is what makes it a 'constant'. In our equation, it is given in units that match up with the other terms, often Joules per mole per Kelvin (J/mol·K). This constant enables us to predict how a hypothetical 'ideal' gas — one without interactions between molecules and volumeless particles — would behave under different temperatures and pressures.

Why Is R 'Ideal'?

In the real world, gases don’t always follow the ideal model, especially at high pressure or low temperature, where their particles do interact and have volume. But R comes from the ideal gas law which assumes ideal behavior, making it an integral part of this model that lets us approach real gas behavior under many conditions.