Question

If a real gas follows equation \(\mathrm{P}(\mathrm{V}-\mathrm{nb})=\mathrm{RT}\) at low pressure, then for a graph between d/P vs. P, (where \(\mathrm{d}\) is the density of gas) (a) Intercept is \(\frac{\mathrm{MR}}{\mathrm{T}}\) (b) Intercept is \(\frac{\mathrm{M}}{\mathrm{RT}}\) (c) Slope is \(-\frac{b}{M(R T)^{2}}\) (d) Slope is \(-\frac{\mathrm{Mb}}{(\mathrm{RT})^{2}}\)

Short answer

The intercept is \( \frac{M}{RT} \) (b) and the slope is \(-\frac{Mb}{(RT)^2}\) (d).

Step-by-step solution

Express the Equation in Terms of Density

The given equation is \( P(V-nb) = RT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( b \) is a correction factor, \( R \) is the universal gas constant, and \( T \) is temperature. Density \( d \) of a gas is defined as \( d = \frac{m}{V} = \frac{Mn}{V} \), where \( M \) is the molar mass. Express \( V \) as \( V = \frac{Mn}{d} \) and substitute into the equation: \[ P\left(\frac{Mn}{d} - nb\right) = RT \].

Simplify the Equation for Pressure

Rearrange the equation to isolate \( P \): \[ P\left(\frac{Mn}{d} - nb\right) = RT \]\[ P\frac{Mn}{d} - Pnb = RT \]\[ P = \frac{RT}{\frac{Mn}{d} - nb} \]\[ P = \frac{RTd}{Mn - nbd} \]Simplify further:\[ d = \frac{MRn}{RT-Pb} \].

Analyze Graph of d/P vs. P

Rearrange the equation to find \( \frac{d}{P} \): \[ d = \frac{MRT}{(RT+Pb)n} \], giving \[\frac{d}{P} = \frac{MR}{RT+Pb} \].Assume low pressure, so \( Pb \) is negligible compared to \( RT \) and approximate:\[ \frac{d}{P} \approx \frac{M}{RT} - \frac{Mb}{(RT)^2} P \].

Identify Intercept and Slope

Identify the intercept as \( \frac{M}{RT} \). Thus, the intercept is option (b). The slope is found by the term associated with \( P \):\(-\frac{Mb}{(RT)^2}\), thus the slope is option (d).

Key concepts

Density of Gases

In the world of gases, density is like their "weight problem." Gases are substances without a defined shape or volume, making them a bit tricky to pin down in your everyday understanding. However, for any gas, density, represented as \(d\), is defined as the mass of the gas per unit of volume. This can be expressed with the formula:

• \(d = \frac{m}{V} = \frac{Mn}{V}\), where \(m\) is the mass, \(M\) is the molar mass, and \(V\) is the volume.

The key thing to remember is that density relates back to how much "stuff" is crammed into a space.
This becomes particularly interesting when dealing with gases, as changing the conditions such as pressure or temperature can cause the gas to expand, contract, and thus, change its density.
Understanding how gases behave under different conditions is essential for making sense of this exercise. In real-life applications, knowing the density of a gas can help in deciding how to store it or how it will behave in natural conditions like the atmosphere. In formulaic terms, when the density of a gas changes under stable conditions, it's usually due to changes in temperature or pressure.

Ideal Gas Law

The ideal gas law is like the rulebook gases follow under ideal conditions. This is a simplified equation of state for gases, and it ties together four very important properties of gases:

• Pressure (\(P\))
• Volume (\(V\))
• Temperature (\(T\))
• Number of moles (\(n\))

This relationship is given by the formula \(PV = nRT\), where \(R\) is the universal gas constant.
The ideal gas law assumes that gas particles do not interact and occupy no space, which isn't always true, yet it's a great starting framework to understand gas behavior.
When the conditions deviate from the "ideal" (i.e., high pressure or low temperature), gases exhibit more significant deviations from this law, thanks to their real interactions and volume overcoming the assumptions in the ideal gas law.
Such observations lead to alternate equations, like the real gas equation used in our exercise: \(P(V-nb) = RT\), which accounts for volume occupied by gas particles. This adjustment is essential for describing gases under non-ideal conditions.

Pressure-Volume Relationship

The pressure-volume relationship is a fundamental concept in understanding how gases interact.
According to Boyle’s Law, under constant temperature, the pressure of a gas is inversely proportional to its volume. This means that if you increase the volume of the gas (imagine stretching a balloon), the pressure drops, because particles are more spread out.In our real gas equation, \(P(V-nb) = RT\), the \(nb\) term accounts for the actual volume occupied by the gas particles themselves, a vital correction at low pressures where volume correction is significant.
The graph of \(d/P\) versus \(P\) from the exercise captures this relationship. As explained, at low pressures, which is our specific interest, \(Pb\) becomes negligible, making the formula approximately \(\frac{d}{P} \approx \frac{M}{RT} - \frac{Mb}{(RT)^2} P\).
This shows the linear dependency where the pressure increases lead to a slight change in the slope, influenced by the real gas behavior adaptations.