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Chapter 14: Problem 5

If for hydrogen \(C_{p}-C_{V}=m\) and for the nitrogen \(C_{p}-C_{V}=n\), where \(C_{p}, C_{V}\) refer to specific heats per unit mass respectively at constant pressure and constant volume, the relation between \(m\) and \(n\) is (a) \(m=14 n\) (b) \(n=7 n\) (c) \(m=7 n\) (d) \(n=14 n\)

Short Answer

Expert verified
The relation is \(m = 14n\), so option (a) is correct.

Step by step solution

01

Understand the Problem

We are given two gases, hydrogen and nitrogen, with the specific heat difference at constant pressure and constant volume defined as \(C_p - C_V = m\) for hydrogen and \(C_p - C_V = n\) for nitrogen. We need to find the relationship between \(m\) and \(n\).
02

Recall the Formula for Ideal Gases

For any ideal gas, the difference in specific heats is given by \(C_p - C_V = R\), where \(R\) is the specific gas constant. Thus, for hydrogen \(C_p - C_V = R_{H_2} = m\) and for nitrogen, \(C_p - C_V = R_{N_2} = n\).
03

Calculate the Gas Constant for Each Gas

The specific gas constant \(R\) is related to the universal gas constant \(R_u\) by \(R = \frac{R_u}{M}\), where \(M\) is the molar mass of the gas. Hydrogen has a molar mass of 2 g/mol and nitrogen has a molar mass of 28 g/mol.
04

Relate m and n Using Molar Masses

Using the relation \(R_{H_2} = \frac{R_u}{2}\) for hydrogen and \(R_{N_2} = \frac{R_u}{28}\) for nitrogen, we find that \(m = \frac{R_u}{2}\) and \(n = \frac{R_u}{28}\). By comparing these, \(m = 14n\).
05

Select the Correct Option

The relationship \(m = 14n\) corresponds to option (a).

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Key Concepts

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

Specific Heat at Constant Pressure
Specific heat at constant pressure, often denoted as \(C_p\), describes the amount of energy required to raise the temperature of a unit mass of a substance by one degree Celsius while maintaining constant pressure. This property is crucial in understanding how gases behave when they are heated in open systems, such as the atmosphere, where the pressure doesn't remain constant.

For an ideal gas, \(C_p\) can be expressed in terms of the universal gas constant \(R_u\) and the molar mass \(M\). This is because the energy needed to increase temperature at constant pressure involves additional work done by the gas expanding against the surrounding pressure.
  • In mathematical terms, \(C_p = C_v + R\), where \(R\) is the specific gas constant.
  • The specific gas constant \(R\) is defined as \(R = \frac{R_u}{M}\).
Understanding \(C_p\) helps predict how energy changes in a gas when it expands and interacts with its environment.
Specific Heat at Constant Volume
Specific heat at constant volume, symbolized by \(C_v\), measures the amount of heat required to increase the temperature of a unit mass of a substance by one degree Celsius while keeping the volume constant. This concept is particularly relevant in closed systems where the volume does not change, such as in piston engines.

In such cases, the gas does not do work on its surroundings, focusing all energy on increasing the internal temperature.
  • For any ideal gas, the relationship \(C_p - C_v = R\) holds true where \(R\) is the specific gas constant.
  • This means that \(C_v\) is a crucial part of quantifying the internal energy change of a gas when isolated from pressure changes.
Ultimately, understanding \(C_v\) describes how gases store energy and how this energy affects temperature under constant volume.
Ideal Gas Law
The Ideal Gas Law is a fundamental equation in chemistry and physics that describes the behavior of an ideal gas. It combines several separate gas laws into one formula: \(PV = nRT\). This equation relates the pressure \(P\), volume \(V\), number of moles \(n\), ideal gas constant \(R\), and temperature \(T\) of an ideal gas.
  • Ideal gases are hypothetical gases that follow this law perfectly. Real gases approximate this behavior at high temperature and low pressure.
  • The constant \(R\) is critical as it links these macroscopic quantities achieved through the Ideal Gas Law. This constant appears in other related thermodynamic equations like \(C_p - C_v = R\).
The Ideal Gas Law assists in calculating unknown properties of a gas when others are known, playing a vital role in both theoretical and practical applications across various fields of science.

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Most popular questions from this chapter

A cylinder with a movable piston contains 3 moles of hydrogen at standard temperature and pressure. The walls of the cylinder are made of heat insulator, and the piston is insulated by having a pile of sand on it/ By what factor does the pressure of the gas increase, if the gas is compressed to half its original volume? [NCERT Exemplar) (a) \(1.40\) (b) \(1.60\) (c) \(2.64\) (d) \(1.94\)

Pressure \(p\), volume \(V\) and temperature \(T\) of a certain material are related by \(p=\alpha T^{2} / V\), where \(\alpha\) is constant. Work done by the material when temperature changes from \(T_{0}\) to \(2 T_{0}\) and pressure remains constant is (a) \(3 \alpha T_{0}^{2}\) (b) \(5 \alpha T_{0}^{2}\) (c) \(\frac{3}{2} \alpha T_{0}^{2}\) (d) \(7 \alpha T_{0}^{2}\)

If an average person jogs, hse produces \(14.5 \times 10^{3} \mathrm{cal} / \mathrm{min}\). This is removed by the evaporation of sweat. The amount of sweat evaporated per minute (assuming \(1 \mathrm{~kg}\) requires \(580 \times 10^{3}\) cal for evaporation) is \(\quad\) [NCERT Exemplar] (a) \(0.25 \mathrm{~kg}\) (b) \(2.25 \mathrm{~kg}\) (c) \(0.05 \mathrm{~kg}\) (d) \(0.20 \mathrm{~kg}\)

When an ideal monoatomic gas is heated at constant pressure, fraction of heat energy supplied which increases the internal energy of gas is (a) \(2 / 5\) (b) \(3 / 5\) (c) \(3 / 7\) (d) \(3 / 4\)

A Carnot engine whose source is at \(400 \mathrm{~K}\) takes \(200 \mathrm{cal}\) of heat and rejects \(150 \mathrm{cal}\) to the sink. What is the temperature of the sink? (a) \(800 \mathrm{~K}\) (b) \(400 \mathrm{~K}\) (c) \(300 \mathrm{~K}\) (d) Cannot say
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