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Chapter 15: Problem 80

The temperature of 2.5 mol of helium (a monatomic gas) is lowered by \(35 \mathrm{~K}\) under conditions of constant volume. Assuming that helium behaves as an ideal gas, how much heat is removed from the gas?

Short Answer

Expert verified
The heat removed is approximately -1091 J.

Step by step solution

01

Understand the Problem

We need to calculate the amount of heat removed from a helium gas, which is a monatomic ideal gas. The change in temperature is given, and the process occurs at constant volume.
02

Identify the Formula

For an ideal gas at constant volume, the heat removed or added can be calculated using the formula: \( q = nC_v\Delta T \), where \( n \) is the number of moles, \( C_v \) is the molar heat capacity at constant volume, and \( \Delta T \) is the change in temperature.
03

Determine Heat Capacity

For a monatomic ideal gas, the molar heat capacity at constant volume \( C_v \) is \( \frac{3}{2} R \), where \( R \) is the ideal gas constant, approximately \( 8.31 \, J/(mol \cdot K) \). Thus, \( C_v = \frac{3}{2} \times 8.31 \, J/(mol \cdot K) \approx 12.47 \, J/(mol \cdot K) \).
04

Calculate the Temperature Change

The temperature change \( \Delta T \) is \( -35 \, K \), as the gas is cooled by 35 K.
05

Plug Values into the Formula

Substitute the known values into the formula: \( q = 2.5 \, mol \times 12.47 \, J/(mol \cdot K) \times (-35 \, K) \).
06

Perform the Calculation

Calculating the heat: \[ q = 2.5 \times 12.47 \times (-35) = -1091.125 \, J \] The negative sign indicates that heat is being removed from the gas.

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Key Concepts

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

Monatomic Gas
In thermodynamics, a monatomic gas is a gas composed of individual atoms, rather than molecules. Helium, for example, is a monatomic gas. This simplicity means each atom is not bonded to others, allowing basic behavior predictions using the ideal gas law.

Monatomic gases possess distinct characteristics. They generally have low density and high kinetic energy relative to their mass.
  • Examples include noble gases such as helium, neon, and argon.
  • Their atoms move freely, resulting in straightforward calculations when applying gas laws.
Understanding these properties is essential when calculating changes like heat removal in thermodynamic problems, allowing accurate use of models such as the ideal gas law.
Heat Capacity
The heat capacity of a substance is critical in determining how much energy it takes to change its temperature. For gases, the specific heat capacity differs if it is measured at constant volume or constant pressure.

For a monatomic ideal gas like helium at constant volume, the molar heat capacity (_C_v_) is expressed as:
  • \( C_v = \frac{3}{2}R \)
  • Where \( R \) is the ideal gas constant, approximately \( 8.31 \, J/(mol \cdot K) \).
This equation allows us to determine how sensitive a gas is to heat changes, an essential factor when predicting the amount of heat removed or added.
Temperature Change
Temperature change (_螖T_) in thermodynamics reveals how much a certain condition鈥攍ike cooling鈥攁ffects a gas's thermal state. For helium, reducing the temperature by \( 35 \, K \) directly influences its internal energy.
  • Temperature change is often calculated in Kelvin, providing a direct line of measurement in scientific problems.
  • In problems where heat is removed, _螖T_ indicates a temperature drop, and hence, a subtraction from internal energy.
Grasping temperature鈥檚 impact allows for accurate applications of energy equations and prediction of system changes in gas dynamics.
Constant Volume Process
In a constant volume process, the gas does not change its volume. This constraint simplifies calculations because volume remains stable, focusing instead on temperature and pressure variables.

The constant volume process affects physical properties and makes gas behavior more predictable in calculations using the ideal gas law.
  • No work is done by the gas since work comes from volume change. Only internal energy alters due to heat exchange.
  • This concept is crucial for calculations involving heat transfer, notably using the formula: \( q = nC_v\Delta T \).
By maintaining volume, calculations focus on how temperature adjustments affect energy transfers, crucial for understanding enthalpy and heat in gas systems.

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Most popular questions from this chapter

A diesel engine does not use spark plugs to ignite the fuel and air in the cylinders. Instead, the temperature required to ignite the fuel occurs because the pistons compress the air in the cylinders. Suppose air at an initial temperature of \(21^{\circ} \mathrm{C}\) is compressed adiabatically to a temperature of \(688^{\circ} \mathrm{C}\). Assume the air to be an ideal gas for which \(\gamma=\frac{7}{5}\). Find the compression ratio, which is the ratio of the initial volume to the final volume.

A refrigerator operates between temperatures of 296 and \(275 \mathrm{~K}\). What would be its maximum coefficient of performance?

Two Camot engines, \(A\) and \(B\), utilize the same hot reservoir, but engine \(\mathrm{A}\) is less efficient than engine \(\mathrm{B}\). (a) Which engine produces more work for a given heat input? (b) Which engine has the lower cold-reservoir temperature? Give your reasoning. Carnot engine A has an efficiency of \(0.60\), and Carnot engine \(\mathrm{B}\) has an efficiency of \(0.80\). Both engines utilize the same hot reservoir, which has a temperature of \(650 \mathrm{~K}\) and delivers \(1200 \mathrm{~J}\) of heat to each engine. Find the magnitude of the work produced by each engine and the temperatures of the cold reservoirs that they use. Check to see that your answers are consistent with your answers to the Concept Questions.

The temperature of \(2.5 \mathrm{~mol}\) of a monatomic ideal gas is \(350 \mathrm{~K}\). The internal energy of this gas is doubled by the addition of heat. How much heat is needed when it is added at (a) constant volume and (b) constant pressure?

The pressure and volume of an ideal monatomic gas change from \(A\) to \(B\) to \(C,\) as the drawing shows. The curved line between \(A\) and \(C\) is an isotherm. (a) Determine the total heat for the process and (b) state whether the flow of heat is into or out of the gas.
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