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Chapter 8: Problem 1112

70 calorie of heat are required to raise the temperature of 2 mole of an ideal gas at constant pressure from \(30^{\circ} \mathrm{C}\) to $35^{\circ} \mathrm{C}$ The amount of heat required to raise the temperature of the same gas through the same range at constant volume is $\ldots \ldots \ldots \ldots \ldots .$ calorie. (A) 50 (B) 30 (C) 70 (D) 90

Short Answer

Expert verified
The heat required to raise the temperature of the same gas through the same range at constant volume is approximately 50 calories. The correct answer is (A) 50.

Step by step solution

01

Identify the given information

We are given that 70 calories of heat are required to raise the temperature of 2 moles of an ideal gas from \(30^{\circ} \mathrm{C}\) to \(35^{\circ} \mathrm{C}\) at constant pressure. Also, we need to find the heat required to raise the temperature of the gas through the same range at constant volume.
02

Use the heat formula at constant pressure

We can use the formula for heat at constant pressure: \[Q_p = nC_p \Delta T\] where \(Q_p\) is the heat at constant pressure, \(n\) is the number of moles, \(C_p\) is the specific heat capacity at constant pressure, and \(\Delta T\) is the change in temperature.
03

Calculate the specific heat capacity at constant pressure

We can plug in the given values and solve for \(C_p\): \[70 \text{ cal} = 2 \text{ moles} \cdot C_p \cdot (35^{\circ} \mathrm{C} - 30^{\circ} \mathrm{C})\] First, calculate the change in temperature: \(\Delta T = 35^{\circ} \mathrm{C} - 30^{\circ} \mathrm{C} = 5^{\circ} \mathrm{C}\) Now, plug it into the equation to find \(C_p\): \[C_p = \frac{70 \text{ cal}}{2 \text{ moles} \cdot 5^{\circ} \mathrm{C}} = 7 \frac{\text{cal}}{\text{mole}\cdot ^{\circ} \mathrm{C}}\]
04

Find the specific heat capacity at constant volume

For an ideal gas, the relationship between specific heat capacities at constant pressure and constant volume is: \[C_p - C_v = R\] where \(R\) is the gas constant. For this problem, we can use the value of \(R = 1.987 \frac{\text{cal}}{\text{mole}\cdot ^{\circ} \mathrm{C}}\). Now, we can find \(C_v\): \[C_v = C_p - R = 7 \frac{\text{cal}}{\text{mole}\cdot ^{\circ} \mathrm{C}} - 1.987 \frac{\text{cal}}{\text{mole}\cdot ^{\circ} \mathrm{C}} = 5.013 \frac{\text{cal}}{\text{mole}\cdot ^{\circ} \mathrm{C}}\]
05

Calculate the heat at constant volume

Now that we have \(C_v\), we can calculate the heat at constant volume using the formula: \[Q_v = nC_v \Delta T\] Plug in the values: \[Q_v = 2 \text{ moles} \cdot 5.013 \frac{\text{cal}}{\text{mole}\cdot ^{\circ} \mathrm{C}} \cdot 5^{\circ} \mathrm{C}\] Calculate the heat at constant volume: \[Q_v = 50.13 \text{ cal}\] Since the given options are all integers, we can round the answer to the nearest integer: \[Q_v \approx 50 \text{ cal}\] So, the heat required to raise the temperature of the same gas through the same range at constant volume is approximately 50 calories. The correct answer is (A) 50.

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Most popular questions from this chapter

The Volume of an ideal gas is 1 liter column and its Pressure is equal to $72 \mathrm{~cm}\( of \)\mathrm{Hg}$. The Volume of gas is made 900 \(\mathrm{cm}^{3}\) by compressing it isothermally. The stress of the gas will be \(\ldots \ldots \ldots \ldots .\) Hg column. (A) \(4 \mathrm{~cm}\) (B) \(6 \mathrm{~cm}\) (C) \(7 \mathrm{~cm}\) (D) \(8 \mathrm{~cm}\)

For an isothermal expansion of a Perfect gas, the value of $(\Delta \mathrm{P} / \mathrm{P})$ is equal to (A) \(-\gamma^{(1 / 2)}\\{(\Delta \mathrm{V}) / \mathrm{V}\\}\) (B) \(-\gamma\\{(\Delta \mathrm{V}) / \mathrm{V}\\}\) (C) \(-\gamma^{2}\\{(\Delta \mathrm{V}) / \mathrm{V}\\}\) \((\mathrm{D})-(\Delta \mathrm{V} / \mathrm{V})\)

Air is filled in a motor tube at \(27^{\circ} \mathrm{C}\) and at a Pressure of 8 atmosphere. The tube suddenly bursts. Then what is the temperature of air. given \(\gamma\) of air \(=1.5\) (A) \(150 \mathrm{~K}\) (B) \(150^{\circ} \mathrm{C}\) (C) \(75 \mathrm{~K}\) (D) \(27.5^{\circ} \mathrm{C}\)

\(1 \mathrm{~mm}^{3}\) Of a gas is compressed at 1 atmospheric pressure and temperature \(27^{\circ} \mathrm{C}\) to \(627^{\circ} \mathrm{C}\) What is the final pressure under adiabatic condition. \(\mathrm{r}=1.5\) (A) \(80 \times 10^{5}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\) (B) \(36 \times 10^{5}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\) (C) \(56 \times 10^{5}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\) (D) \(27 \times 10^{5}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\)

A gas mixture consists of 2 mole of oxygen and 4 mole of argon at temperature \(\mathrm{T}\). Neglecting all vibrational modes, the total internal energy of the system is (A) \(11 \mathrm{RT}\) (B) \(9 \mathrm{RT}\) (C) \(15 \mathrm{RT}\) (D) \(4 \mathrm{RT}\)
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