Ch 19: Work, Heat, and the First Law of Thermodynamics

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 19: Work, Heat, and the First Law of Thermodynamics

Problem 28b

Chapter 19, Problem 28b

A container holds 1.0 g of oxygen at a pressure of 8.0 atm. How much will the temperature increase if this amount of heat energy is transferred to the gas at constant volume?

Verified step by step guidance

Step 1: Identify the given values and the formula to use. The problem involves heat transfer at constant volume, so we use the formula for heat transfer in an ideal gas:

Q = nC_vΔT

, where

Q

is the heat energy transferred,

n

is the number of moles of gas,

C_v

is the molar heat capacity at constant volume, and

ΔT

is the temperature change.

Step 2: Calculate the number of moles of oxygen gas. Use the molar mass of oxygen, which is approximately 32 g/mol. The number of moles is given by

n = rac{m}{M}

, where

m

is the mass of the gas (1.0 g) and

M

is the molar mass.

Step 3: Determine the molar heat capacity at constant volume,

C_v

. For diatomic gases like oxygen,

C_v

is approximately

rac{5}{2}R

, where

R

is the universal gas constant (8.314 J/(mol·K)).

Step 4: Rearrange the heat transfer formula to solve for the temperature change:

ΔT = rac{Q}{nC_v}

. Substitute the values of

Q

n

, and

C_v

into the equation.

Step 5: Perform the calculation to find

ΔT

. This will give the increase in temperature of the gas in Kelvin. Ensure the units are consistent throughout the calculation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. This law is fundamental in understanding how gases behave under various conditions, particularly when analyzing changes in temperature and pressure while keeping volume constant.

Heat Transfer and Internal Energy

Heat transfer refers to the energy exchanged between a system and its surroundings due to a temperature difference. In the context of gases, when heat is added at constant volume, it increases the internal energy of the gas, which is directly related to its temperature, as described by the first law of thermodynamics.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius. For gases, this concept is crucial when calculating temperature changes resulting from heat transfer, as it determines how much the temperature will increase based on the amount of heat added.

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