Question

A container holds a mixture of three nonreacting gases: \(n_{1}\) moles of the first gas with molar specific heat at constant volume \(C_{1}\), and so on. Find the molar specific heat at constant volume of the mixture, in terms of the molar specific heats and quantities of the three separate gases.

Short answer

The molar specific heat at constant volume of the mixture is given by \[C_{mixture} = \frac{n_{1}C_{1} + n_{2}C_{2} + n_{3}C_{3}}{n_{1} + n_{2} + n_{3}}\]

Step-by-step solution

Understand the Molar Specific Heat at Constant Volume

Molar specific heat at constant volume, denoted by C, is the amount of heat required to raise the temperature of 1 mole of the substance by 1 degree at constant volume. Hence, for any mixture, the total heat capacity would be the sum of the heats required to raise the temperature of each component of the mixture.

Determine the Total Number of Moles

First, find the total number of moles in the mixture. This can be done by adding together the amounts of each of the three gases, i.e., \(n_{total} = n_{1} + n_{2} + n_{3}\).

Calculate the Total Heat Capacity

Next, calculate the total heat capacity of the mixture. This is equal to the sum of the heat capacities of each component of the mixture. Heat capacity at constant volume for a substance is the product of the number of moles and the molar specific heat. Therefore, total heat capacity, \(C_{total}\) is given by \[C_{total} = n_{1}C_{1} + n_{2}C_{2} + n_{3}C_{3}\]

Calculate the Molar Specific Heat of the Mixture

The molar specific heat at constant volume of the mixture \(C_{mixture}\) is then given by the total heat capacity divided by the total number of moles, which in formulas is expressed as: \[C_{mixture} = \frac{C_{total}}{n_{total}} = \frac{n_{1}C_{1} + n_{2}C_{2} + n_{3}C_{3}}{n_{1} + n_{2} + n_{3}}\]

Key concepts

Understanding Thermodynamics

Thermodynamics is a branch of physics that deals with the relationships between heat and other forms of energy. In the context of our exercise, thermodynamics governs the principles behind the molar specific heat at constant volume. This concept is pivotal because it dictates how energy, in the form of heat, is transferred to and from objects to change their temperature under specific conditions, like constant volume in our scenario.

When discussing gas mixtures, thermodynamics becomes especially interesting as it allows us to understand the behavior of each gas component and predict the overall heat capacity of the mixture. The laws of thermodynamics also ensure that the energy conservation principle is applied, ensuring that the calculated molar specific heat at constant volume for the mixture is consistent with the sum of the components' contributions.

Diving into Heat Capacity

Heat capacity is one of the central concepts in thermodynamics as it measures the amount of heat energy required to change the temperature of a substance. The molar specific heat at constant volume, specifically, is the heat capacity per mole of a substance when its volume is held constant. This focus on per mole basis allows for the ease of comparison between materials and is crucial in calculations involving gas mixtures.

The exercise provided focuses on finding the average molar specific heat of a gas mixture. Here, understanding the individual heat capacities of the various gases contributes cumulatively to understanding the mixture's overall capacity to absorb heat. This understanding is important in a wide range of applications, from industrial processes to environmental science.

Analyzing Gas Mixtures

Gas mixtures are commonly found both in natural settings and in technological applications. The behavior of these mixtures can be complex due to the interactions between different gases, each with its own properties and reactions to temperature changes. The molar specific heat at constant volume for a gas mixture is an average value that represents the heat capacity of the mixture as if it were a single substance.

The step-by-step solution involves summing the products of the number of moles and the specific heats of the individual gases to find the total heat capacity. From this, the molar specific heat of the mixture is calculated by dividing by the total number of moles. This simplification is highly useful in engineering and science as it can dramatically simplify calculations involving the thermal properties of gas mixtures.

Exploring Molecular Physics

Molecular physics focuses on the physical properties of molecules, the chemical bonds between atoms, and the molecular dynamics that dictate how substances behave. When it comes to molar specific heat, molecular physics helps explain why different gases have unique values. It's all about the molecular structure and degrees of freedom—rotational, translational, and vibrational motions that molecules can undergo.

The molar specific heat relates directly to these degrees of freedom, which is why monoatomic gases like helium have different specific heats than diatomic gases such as nitrogen. Understanding the molecular nature of the gases in the mixture allows us to see why the averaged molar specific heat at constant volume doesn't simply equal the arithmetic mean of the individual specific heats; instead, it's a weighted average based on the number of moles and the type of gas molecules present.