Question

A vessel contains 1 mole of \(\mathrm{O}_{2}\) gas at a temperature \(T\). The pressure of the gas is \(P\). An identical vessel containing 1 mole of \(\mathrm{He}\) gas at a temperature \(2 T\) has a pressure of (a) \(P / 8\) (b) \(P\) (c) \(2 P\) (d) \(8 P\)

Short answer

The pressure of 1 mole of He gas at temperature 2T is (c) 2P.

Step-by-step solution

Understand the Ideal Gas Law

The ideal gas law can be stated as PV = nRT where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the temperature. Since we are dealing with ideal gases and the volumes and the number of moles in both cases are the same, we can say that the pressure of a gas is directly proportional to its temperature (P/T = constant).

Relate the Pressure and Temperature of O2

Since the pressure of O2 is P at temperature T, we can write the relationship as P1/T1 for O2.

Relate the Pressure and Temperature of He

For He gas at temperature 2T, let's denote the pressure it exerts as P2. Applying the ideal gas law, and since we are looking for how P2 relates to P (the pressure of O2 at T), we can write the relationship as P2/T2 for He, where T2 = 2T.

Compare Ratios to Find the Pressure of He Gas

Since the pressure to temperature ratio is constant, set the ratio for O2 equal to the ratio for He to find the pressure P2. Using P1/T1 = P2/T2, we substitute P1 with P and T1 with T. We then substitute T2 with 2T and solve for P2: P/T = P2/(2T). Simplifying this, we get P = P2/2, so P2 = 2P.

Key concepts

Gas Laws and Temperature-Pressure Relationship

Understanding the relationship between temperature and pressure in gases is essential when examining their behaviors, especially under varying conditions. This correlation is part of the gas laws that describe how these variables interplay.

The correlation between temperature and pressure within a gas is anchored on one of the basic principles of kinetic theory, which suggests that as the kinetic energy of the gas particles increases due to an increase in temperature, the particles move more rapidly and collide against the container walls more frequently and with greater force. This heightened movement escalates the pressure exerted by the gas.

In mathematical terms, this is expressed as a direct relationship, provided the volume of the gas and the number of gas particles remain constant. This means that if you double the temperature of a gas (measured in Kelvin), maintaining the same volume and amount of gas, the pressure will also double. Conversely, if the temperature is halved, the pressure will reduce by half. This relationship is essential when you're working with problems regarding gas behavior under different temperatures.

Ideal Gas Equation

The ideal gas equation is a fundamental tool in understanding the properties of gases. The equation itself is represented as \( PV = nRT \), where \( P \) stands for pressure, \( V \) for volume, \( n \) for the number of moles of the gas, \( R \) for the ideal gas constant, and \( T \) for the absolute temperature.

This equation is vital because it relates the four critical physical quantities that define the state of an ideal gas. For an ideal gas, we assume no intermolecular forces and that the volume of the individual gas particles is negligible compared to the container's volume. The gas constant \( R \) is the same for all gases and provides a bridge between the laboratory and universal gas properties.

Significance of Each Variable

• \( P \) (Pressure): It's a measure of the force that the gas exerts on its container per unit area.
• \( V \) (Volume): Represents the space that the gas occupies.
• \( n \) (Moles): Reflects the amount of substance or the quantity of molecules present in a gas.
• \( T \) (Absolute Temperature): It's crucial to measure it in Kelvin, as the equation relies on an absolute scale to be accurate.

For problem-solving, mastering the ideal gas equation enables prediction of one property if the others are known, allowing for better comprehension of gas behavior under different conditions.

Comparative Gas Pressure Calculations

Comparative gas pressure calculations are critical when analyzing the behavior of gases under varying conditions. They typically involve using the ideal gas law to assess how a change in one or more properties of a gas affects another property, given the gas follows ideal behavior.

Given that ideal gas behavior is assumed, and when comparing two scenarios that have the same volume and number of moles of gases, we can simplify the ideal gas equation to a direct relationship of \( P/T = \text{constant} \). This implies that the ratio of pressure to temperature for one state of the gas can be equated to the ratio of pressure to temperature for another state.

When given a problem where you have to compare the pressures of two different gases at different temperatures—like in our exercise—it's important to note that as long as the gas is ideal and the number of moles and volume stay constant, only the temperatures are responsible for the differences in pressures. This makes for a straightforward calculation to determine how the pressure will increase or decrease with the change in temperature.