Question

A mixture containing \(0.477 \mathrm{~mol} \mathrm{He}(\mathrm{g}), 0.280 \mathrm{~mol} \mathrm{Ne}(\mathrm{g})\), and \(0.110 \mathrm{~mol} \mathrm{Ar}(g)\) is confined in a \(7.00\) -L vessel at \(25^{\circ} \mathrm{C}\). (a) Calculate the partial pressure of each of the gases in the mixture. (b) Calculate the total pressure of the mixture.

Short answer

(a) The partial pressures of the gases are: \(P_{He} = 1.634 \mathrm{~atm}\), \(P_{Ne} = 0.962 \mathrm{~atm}\), and \(P_{Ar} = 0.378 \mathrm{~atm}\). (b) The total pressure of the mixture is \(P_{total} = 2.974 \mathrm{~atm}\).

Step-by-step solution

Convert the temperature from Celsius to Kelvin

First, we need to convert the temperature from Celsius to Kelvin using the following formula: \(T(K) = T(°C) + 273.15\) So, for this problem, the temperature is: \(T(K) = 25°C + 273.15 = 298.15 K\)

Calculate the partial pressure of each gas using the Ideal Gas Law

Now we can use the Ideal Gas Law to find the partial pressure of each gas. We'll use the given number of moles for each gas, the volume of the vessel (7.00 L), and the temperature we just calculated (298.15 K). For He: \(P_{He}V = n_{He}RT\) \(P_{He} = \frac{n_{He}RT}{V}\) \(P_{He} = \frac{(0.477 \mathrm{~mol})(0.0821 \frac{L \cdot atm}{mol \cdot K})(298.15 K)}{(7.00 L)}\) \(P_{He} = 1.634 \mathrm{~atm}\) For Ne: \(P_{Ne}V = n_{Ne}RT\) \(P_{Ne} = \frac{n_{Ne}RT}{V}\) \(P_{Ne} = \frac{(0.280 \mathrm{~mol})(0.0821 \frac{L \cdot atm}{mol \cdot K})(298.15 K)}{(7.00 L)}\) \(P_{Ne} = 0.962 \mathrm{~atm}\) For Ar: \(P_{Ar}V = n_{Ar}RT\) \(P_{Ar} = \frac{n_{Ar}RT}{V}\) \(P_{Ar} = \frac{(0.110 \mathrm{~mol})(0.0821 \frac{L \cdot atm}{mol \cdot K})(298.15 K)}{(7.00 L)}\) \(P_{Ar} = 0.378 \mathrm{~atm}\) (a) So the partial pressure of each gas is: \(P_{He} = 1.634 \mathrm{~atm}\) \(P_{Ne} = 0.962 \mathrm{~atm}\) \(P_{Ar} = 0.378 \mathrm{~atm}\)

Calculate the total pressure of the mixture

Now we can add up the partial pressures of each gas to find the total pressure: \(P_{total} = P_{He} + P_{Ne} + P_{Ar}\) \(P_{total} = 1.634 \mathrm{~atm} + 0.962 \mathrm{~atm} + 0.378 \mathrm{~atm}\) \(P_{total} = 2.974 \mathrm{~atm}\) (b) The total pressure of the mixture is \(P_{total} = 2.974 \mathrm{~atm}\).

Key concepts

Ideal Gas Law

The Ideal Gas Law is a crucial equation in chemistry and physics that relates the pressure, volume, temperature, and number of moles of a gas. Expressed as PV = nRT, it allows us to calculate any one of these four properties when the other three are known. In this relation, P stands for pressure in atmospheres (atm), V is volume in liters (L), n represents the number of moles of the gas, R is the ideal gas constant which is approximately 0.0821 L·atm/mol·K, and T is the temperature in Kelvin (K).

A critical aspect of applying the Ideal Gas Law is ensuring that temperature is appropriately converted to Kelvin, as it is a standard condition for the equation. When working with gas mixtures, the law helps in determining the partial pressure of each gas, contributing to the overall understanding of gas behavior under different conditions. It's a foundational concept that serves as a gateway to exploring more advanced gas laws and their applications.

Gas Mixture

A gas mixture involves more than one gas combined in a single container, with each gas exerting its own partial pressure. The concept of partial pressure is essential when dealing with mixtures, as it refers to the pressure a single gas in the mixture would exert if it occupied the entire volume of the container on its own.

To calculate the partial pressure of each component in a mixture, we apply the Ideal Gas Law for each gas separately, while maintaining the same temperature and volume for all gases in the mixture. Then, Dalton's Law of Partial Pressures allows us to add up these individual pressures to obtain the total pressure exerted by the mixture. In the given exercise, understanding the behavior of the gas mixture was crucial in determining the pressure contributed by each type of gas—helium (He), neon (Ne), and argon (Ar).

Kelvin Temperature Conversion

Temperature plays a fundamental role in gas behavior and must be accurately measured. The Kelvin scale is the standard unit of temperature in the physical sciences, and it starts from absolute zero, the theoretical point where particles have minimal thermal motion.

To convert Celsius to Kelvin, which is necessary for using the Ideal Gas Law, the formula T(K) = T(°C) + 273.15 is used. This adjustment ensures that the temperature is absolute, which is a requirement for calculations related to gas properties. In educational content, emphasizing the importance of this conversion helps students avoid common mistakes and understand the universality of the Kelvin scale in scientific calculations.

Molar Volume

Molar volume is another key concept when working with gases. It is defined as the volume occupied by one mole of a substance. For gases, a key reference point is the molar volume at standard temperature and pressure (STP), which is 22.4 L for one mole of any ideal gas.

In the context of the exercise, while molar volume wasn't a direct topic, it implicitly set the stage for calculations regarding the volume of gas mixtures. Given the volume of the vessel and the amount of each gas present in the mixture, the Ideal Gas Law allows for a clear understanding of how much space one mole of gas occupies. It is particularly interesting to note that real gases have molar volumes that can deviate from the ideal depending on conditions, which is why the Ideal Gas Law is a simplified model that works best under certain conditions.