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Chapter 21: Problem 23

A container has a mixture of two gases: \(n_{1}\) mol of gas 1 having molar specific heat \(C_{1}\) and \(n_{2}\) mol of gas 2 of molar specific heat \(C_{2} .\) (a) Find the molar specific heat of the mixture. (b) What If? What is the molar specific heat if the mixture has \(m\) gases in the amounts \(n_{1}, n_{2}, n_{3}, \ldots, n_{m}\) with molar specific heats \(C_{1}, C_{2}, C_{3}, \ldots, C_{m},\) respectively?

Short Answer

Expert verified
The molar specific heat of the mixture of two gases can be calculated using the formula \(C = (n_{1}C_{1} + n_{2}C_{2}) / (n_{1} + n_{2})\) and generalized for m number of gases with the formula \(C = \frac {\sum _{i=1}^m n_{i}C_{i}}{\sum _{i=1}^m n_{i}}\). The actual values will depend on the specific numbers provided.

Step by step solution

01

Understand the formula to compute the molar specific heat of a mixture comprising of two gases

The molar specific heat, C of a mixture of two gasses is computed as the weighted average of the molar specific heats of the individual gases. This would be given by \(C = (n_{1}C_{1} + n_{2}C_{2}) / (n_{1} + n_{2})\).
02

Applying the formula to find the molar specific heat of the mixtures of the given gases

To find the molar specific heat of the mixture, substitute the values of \(n_{1}\), \(n_{2}\), \(C_{1}\), and \(C_{2}\) into the formula. Let鈥檚 assume these values are given.
03

Calculation for a mixture of many different gases

For the case of 'm' different gases, the formula can be generalized. The molar specific heat of the mixture would be given by \(C = \frac {\sum _{i=1}^m n_{i}C_{i}}{\sum _{i=1}^m n_{i}}\), where \(n_{i}\) is the number of moles of the i-th gas. , and \(C_{i}\) is the molar specific heat of the i-th gas. Assume the values are given, substitute them and calculate.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Thermodynamics
Thermodynamics is the science of energy transfer and energy transformation. It governs the principles of how heat, work, and internal energy within a system, like a container of gas, interact.
In the case of gases, we often deal with concepts like pressure, volume, temperature, and energy. These variables are interrelated, and their interactions dictate how a gas behaves. When gases mix, thermodynamically, they interact without losing their individual properties unless chemically reacted.
Understanding thermodynamics helps us predict how a system will react when subjected to certain conditions, like heating or compression.
  • It provides equations that relate the macroscopic properties of gases.
  • Thermodynamics, by defining specific heats, tells how heat capacities vary based on the constraints applied (constant volume or constant pressure).
If you dive deeper鈥攁 key aspect of thermodynamics dealing with gases is understanding how energy is distributed. This includes the concept of enthalpy and entropy, which are more advanced topics.
Mixture of Gases
A mixture of gases involves combining two or more gases in a given volume without them reacting chemically. Each gas in the mixture retains its properties.
The question often arises about how to calculate properties like pressure, temperature, and heat capacities for the mixture, as seen in the problem with molar specific heat.
When dealing with mixtures, each component of the gas contributes to the overall properties based on its proportion in the mixture. The combined effect can be expressed using weighted averages, as is the case with computing molar specific heat here.
  • The total pressure of a mixture in a volume is the sum of partial pressures of the individual gases.
  • The contribution of each gas is calculated based on its proportion and its specific properties.
For instance, the formula utilized for finding molar specific heat: if each gas has a unique molar specific heat, each contributes proportionally to the total based on its mole fraction.
Specific Heat Capacity
Specific heat capacity is a measure of the amount of heat required to change the temperature of a substance by one degree. For gases, this property differs depending on whether the process occurs at constant pressure ( C_p ) or constant volume ( C_v ).
The molar specific heat is focused on this property per mole of substance, which is crucial for analyzing gas mixtures.
Understanding molar specific heat is essential in thermodynamics for energy management. When calculating the total heat capacity of a mixture, you need to know the molar specific heats of individual gases and their mole ratios.
  • A higher molar specific heat indicates a substance needs more energy to increase its temperature.
  • When mixing gases, as in the problem, we determine the molar specific heat of the mixture using the weighted average of the specific heats of the components.
This concept helps in deriving results for multi-component systems, making it indispensable for both theoretical and practical thermodynamics.

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Most popular questions from this chapter

Model air as a diatomic ideal gas with \(M=28.9 \mathrm{g} / \mathrm{mol} .\) A cylinder with a piston contains \(1.20 \mathrm{kg}\) of air at \(25.0^{\circ} \mathrm{C}\) and \(200 \mathrm{kPa} .\) Energy is transferred by heat into the system as it is allowed to expand, with the pressure rising to \(400 \mathrm{kPa}\) Throughout the expansion, the relationship between pressure and volume is given by $$P=C V^{1 / 2}$$ where \(C\) is a constant. (a) Find the initial volume. (b) Find the final volume. (c) Find the final temperature. (d) Find the work done on the air. (e) Find the energy transferred by heat.

A vessel contains \(1.00 \times 10^{4}\) oxygen molecules at \(500 \mathrm{K}\) (a) Make an accurate graph of the Maxwell-Boltzmann speed distribution function versus speed with points at speed intervals of \(100 \mathrm{m} / \mathrm{s}\). (b) Determine the most probable speed from this graph. (c) Calculate the average and rms speeds for the molecules and label these points on your graph. (d) From the graph, estimate the fraction of molecules with speeds in the range \(300 \mathrm{m} / \mathrm{s}\) to \(600 \mathrm{m} / \mathrm{s}\)

Assume that the Earth's atmosphere has a uniform temperature of \(20^{\circ} \mathrm{C}\) and uniform composition, with an effective molar mass of \(28.9 \mathrm{g} / \mathrm{mol}\). (a) Show that the number density of molecules depends on height according to $$n_{V}(y)=n_{0} e^{-m g y / k_{\mathrm{B}} T}$$ where \(n_{0}\) is the number density at sea level, where \(y=0\) This result is called the law of atmospheres. (b) Commercial jetliners typically cruise at an altitude of \(11.0 \mathrm{km} .\) Find the ratio of the atmospheric density there to the density at sea level.

A 1 -L. Thermos bottle is full of tea at \(90^{\circ} \mathrm{C}\). You pour out one cup and immediately screw the stopper back on. Make an order-of-magnitude estimate of the change in temperature of the tea remaining in the flask that results from the admission of air at room temperature. State the quantities you take as data and the values you measure or estimate for them.

At what temperature would the average speed of helium atoms equal (a) the escape speed from Earth, \(1.12 \times 10^{4} \mathrm{m} / \mathrm{s}\) and \((\mathrm{b})\) the escape speed from the Moon, \(2.37 \times 10^{3} \mathrm{m} / \mathrm{s} ?\) (See Chapter 13 for a discussion of escape speed, and note that the mass of a helium atom is \(\left.6.64 \times 10^{-27} \mathrm{kg} .\right)\)
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