Problem 132 - Chemistry: An Atoms-Focused Approach

Question

Derive an equation that expresses the ratio of the densities \(\left(d_{1} \text { and } d_{2}\right)\) of a gas under two different combinations of temperature and pressure: \(\left(T_{1}, P_{1}\right)\) and \(\left(T_{2}, P_{2}\right)\).

Step-by-Step Solution

Answer

Answer: The ratio of the densities is given by the equation d1 / d2 = (P1 * T2) / (T1 * P2), where d1 and d2 are the densities under conditions (T1, P1) and (T2, P2) respectively.

Step 1: Recall the Ideal Gas Law

The Ideal Gas Law is given by: PV = nRT where: P = pressure (in pascals or atm) V = volume (in liters or m3) n = moles of gas R = ideal gas constant T = temperature (in Kelvin)

Step 2: Express Ideal Gas Law in terms of density

Density (d) is defined as mass (m) divided by volume (V), or: d = m / V Since the mass can be expressed as the product of moles (n) and molar mass (M), we have: d = nM / V Rearranging this equation in terms of n/V: n/V = d/M Now, substitute n/V with d/M into the Ideal Gas Law: P = (d/M) RT Then, isolate d: d = PM / RT

Step 3: Find the ratio of densities under both conditions

Now, we have the expression for density in terms of pressure, molar mass, gas constant, and temperature. We can find the ratio of densities under both conditions (T1, P1) and (T2, P2) by dividing their corresponding equations: d1 / d2 = [(P1M1) / (R * T1)] / [(P2M2) / (R * T2)]

Step 4: Simplify the equation

In order to simplify this equation, we will cancel out the terms that are common in both the numerator and the denominator: d1 / d2 = (P1M1 * R * T2) / (R * T1 * P2 * M2) Here, you will observe that R, the gas constant, can be cancelled out directly, as it appears in both the numerator and the denominator. Also, if we assume that both processes involve the same gas, then M1 and M2 would be equal. In that case, M1/M2 can be removed from the equation too: d1 / d2 = (P1 * T2) / (T1 * P2) This is our final simplified equation that expresses the ratio of the densities of a gas under two different combinations of temperature and pressure.

Key Concepts

Density

Density is a measure of how much mass is contained in a given volume. For gases, density can change based on temperature and pressure. The formula to calculate density from the Ideal Gas Law is:

• \( d = \frac{PM}{RT} \)

Here,

• \( d \) is the density,
• \( P \) is the pressure,
• \( M \) is the molar mass of the gas,
• \( R \) is the ideal gas constant,
• \( T \) is the temperature in Kelvin.

The density of a gas is directly proportional to pressure and molar mass but inversely proportional to temperature. When pressure increases, density also increases. Conversely, if temperature rises, density will decrease. This relationship is crucial when deriving the ratio of densities under different conditions, as seen in the exercise.

Pressure

Pressure is a force exerted by the gas particles when they collide with the walls of their container. It's an important concept in the Ideal Gas Law, which links pressure with volume, temperature, and the number of moles.
Different units can measure pressure like pascals, atmospheres, or torr.
In the context of the Ideal Gas Law, represented as \( PV = nRT \), pressure (

• \( P \)

) plays a significant role in determining gas behavior under various conditions.

• Higher pressure implies that gas particles are more densely packed, which can increase the density.
• Conversely, lowering the pressure allows the gas to expand and become less dense.

In calculating ratios of densities at different pressure and temperature settings, pressure influences the numerator and denominator of the density ratio equation and is pivotal for understanding changes in gas characteristics.

Temperature

Temperature reflects the kinetic energy of gas particles. In the Ideal Gas Law formula, temperature is measured in Kelvin, an absolute scale ensuring no negative temperature values. According to the law,

• \( T \) is directly proportional to volume when pressure is held constant.
• As the temperature increases, gas particles move more rapidly, which can cause expansion.

When calculating the gas density,

• density \( (d) \) decreases with an increase in temperature, because the volume expands making particles less densely packed.
This inverse relationship means that temperature has a key influence on the behavior of gases.
In the derived equation for the ratio of densities, temperature terms \( T_1 \) and \( T_2 \) help determine how much the density will change under varying thermal conditions, showing us that higher temperatures generally mean lower densities for gases.