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Chapter 11: Problem 74

If \(Q\) amount of heat is required to increase the temperature of one mole of a monoatomic gas at constant pressure by \(\wedge T\), the amount of heat required to increase the temperature of half mole of a diatomic gas at constant volume by \(2 \wedge T\) will be (a) \(Q\) (b) \(5 / 3 Q\) (c) \(8 Q\) (d) \(3 / 5 Q\)

Short Answer

Expert verified
(a) Q

Step by step solution

01

Understand the Heat Formula

For a monoatomic gas at constant pressure, the amount of heat required is given by \( Q = n C_p \Delta T \), where \( C_p \) for a monoatomic gas is \( \frac{5}{2} R \) and \( n = 1 \) mole. Thus, \( Q = \left(1 \right) \left( \frac{5}{2} R \right) \Delta T = \frac{5}{2} R \Delta T \).
02

Review the Goal

We need to find the amount of heat required for a half mole of a diatomic gas at constant volume and a temperature change of \( 2 \Delta T \).
03

Monoatomic Gas Heat Required

From Step 1, \( Q = \frac{5}{2} R \Delta T \) for one mole of a monoatomic gas at constant pressure.
04

Calculate Heat for Diatomic Gas

For diatomic gas at constant volume, the heat required is given by \( Q' = n C_v \Delta T \). Here, \( C_v \) for a diatomic gas is \( \frac{5}{2} R \), and \( n = \frac{1}{2} \) with \( \Delta T = 2 \Delta T \). So, \( Q' = \frac{1}{2} \times \frac{5}{2} R \times 2 \Delta T = \frac{5}{2} R \Delta T \).
05

Ratio of Heat Required

Compare the heat required: \( Q' = \frac{5}{2} R \Delta T \) is equal to the original heat \( Q = \frac{5}{2} R \Delta T \). Hence, the ratio of the heat required for the diatomic gas to monoatomic gas is \( Q' = Q \).
06

Conclusion

The heat required to increase the temperature of half mole of diatomic gas at constant volume by \( 2 \Delta T \) is \( Q \). Therefore, the correct answer is option (a) \( Q \).

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Key Concepts

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

Heat Capacity
Heat capacity is an essential concept in thermodynamics, defined as the amount of heat required to change the temperature of a substance by a certain amount. It is typically expressed in units of joules per degree Celsius (J/掳C). There are two main types of heat capacities relevant to gases:

  • Specific Heat Capacity at Constant Volume ( C_v): Refers to the heat capacity when the volume is kept constant. In this scenario, the gas does not perform any work on its surroundings.
  • Specific Heat Capacity at Constant Pressure ( C_p): Refers to the heat capacity when the pressure is constant, which is usually greater than C_v for a given gas since the gas does work on its surroundings as it expands.
Understanding these capacities is crucial because they determine how much energy a substance can hold and transfer, impacting reactions and physical processes.
Monoatomic Gas
A monoatomic gas is composed of single atoms, typical examples being the noble gases like helium, neon, and argon. They have unique heat capacities:
  • At constant volume, the heat capacity, C_v, for a monoatomic gas, is \( \frac{3}{2} R \).
  • At constant pressure, the heat capacity, C_p, is \( \frac{5}{2} R \).
The relation \( C_p - C_v = R \) often follows because the work done by an expanding gas is equal to the gas constant, R. These values play a crucial role when calculating the energy required for temperature changes in monoatomic gases. The simplicity of monoatomic gases makes them ideal for understanding basic thermodynamic principles.
Diatomic Gas
Diatomic gases consist of molecules made up of two atoms, such as oxygen (O鈧�) and nitrogen (N鈧�). These gases have more complex heat capacities due to additional degrees of freedom that involve rotation and vibrational motion:
  • At constant volume, C_v is \( \frac{5}{2} R \).
  • At constant pressure, C_p is \( \frac{7}{2} R \).
These increased capacities reflect the additional energy storage mechanisms possible in diatomic gases. This complexity means that more energy is required to achieve the same temperature change compared to monoatomic gases, impacting processes where heat and work interplay.
Constant Pressure
When dealing with thermodynamics, the concept of constant pressure is critical for understanding heat transfer in gases. At constant pressure, a gas is allowed to expand or contract, therefore doing work on or being worked on by its surroundings. This leads to:
  • A higher heat capacity ( C_p) compared to constant volume due to work done against external pressure.
  • More heat being required to increase the gas's temperature for a given amount.
Heating a gas at constant pressure involves both increasing its internal energy and providing energy for expansion. This makes it a fundamental part of thermodynamic processes like isobaric processes in engines and chemical reactions.
Constant Volume
In thermodynamics, analyzing processes at constant volume simplifies calculations because the gas cannot expand or contract. At constant volume:
  • The heat capacity ( C_v) does not include work done by or on the gas, focusing purely on changes in internal energy.
  • Less heat is required to change the temperature compared to a constant pressure scenario.
Studying gases at constant volume, such as in rigid containers, gives important insights into changes in internal energy without the variables introduced by volume change. This is particularly relevant in controlled laboratory reactions and confined systems.

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Most popular questions from this chapter

A gas at a pressure \(6 \times 10^{5} \mathrm{Nm}^{-2}\) and volume \(1 \mathrm{~m}^{3}\) expands to \(3 \mathrm{~m}^{3}\) and its pressure changes to \(4 \times 10^{5} \mathrm{Nm}^{-2}\). Given that the indicator diagram is a straight line, work done by the system is (a) \(10 \times 10^{5} \mathrm{~J}\) (b) \(20 \times 10^{5} \mathrm{~J}\) (c) \(5 \times 10^{5} \mathrm{~J}\) (d) \(8 \times 10^{5} \mathrm{~J}\)

Which of the following thermomenters will you prefer to measure temperalures below \(1 \mathrm{~K}\) ? (a) Gas thermomcters (b) Thermoelecuric thermometer (c) Magnetic thermometers (d) Vapour pressure thermomewrs

An ideal gas in a cylinder cxpands according to the relation \(P V^{n}=\) constanl. For the process i (a) \(n>1\) then internal cnergy incrcases (b) \(n>1\) then internal cnergy decreases (c) \(n=0\) then internal cnergy incroases (d) \(n=\gamma\) then internal cnorgy remain constant

Mercury boils at \(367{ }^{\circ} \mathbf{C}\). However, mercury thermometers are made such that they can measure temperature upto \(500^{\circ} \mathrm{C}\). This is done by (a) maintaining vacuum above mercury column in the stem of the thermometer (b) filling nitrogen gas al high pressure above the mercury column (c) filling oxygen gas at high pressure above the mercury column (d) filling nitrogcn gas at low pressure above the mercury column

A disc of radius \(R\) has a hole of radius \(r\) in it. The disc is heated then (a) \(R\) and \(r\) both increase (b) \(R\) increases, \(r\) decreases (c) \(R\) decreases, \(r\) increases (d) \(\mathrm{R}\) and \(r\) both will decrease
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