Ch 20: The Micro/Macro Connection

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 20: The Micro/Macro Connection

Problem 60c

Chapter 20, Problem 60c

2.0 g of helium at an initial temperature of 300 K interacts thermally with 8.0 g of oxygen at an initial temperature of 600 K. How much heat energy is transferred, and in which direction?

Verified step by step guidance

Step 1: Identify the specific heat capacities of helium and oxygen. Helium is a monatomic gas, so its molar specific heat at constant volume is \( C_v = \frac{3}{2}R \), where \( R \) is the universal gas constant. Oxygen is a diatomic gas, so its molar specific heat at constant volume is \( C_v = \frac{5}{2}R \).

Step 2: Calculate the number of moles of helium and oxygen. Use the formula \( n = \frac{m}{M} \), where \( m \) is the mass of the gas and \( M \) is its molar mass. For helium, \( M = 4 \ \text{g/mol} \), and for oxygen, \( M = 32 \ \text{g/mol} \).

Step 3: Determine the heat capacities of helium and oxygen. Multiply the number of moles of each gas by their respective molar specific heat capacities. For helium, \( C_{He} = n_{He} \cdot C_v \), and for oxygen, \( C_{O_2} = n_{O_2} \cdot C_v \).

Step 4: Calculate the final equilibrium temperature \( T_f \) using the principle of conservation of energy. The heat lost by oxygen equals the heat gained by helium: \( C_{O_2} (T_{O_2, \text{initial}} - T_f) = C_{He} (T_f - T_{He, \text{initial}}) \). Solve for \( T_f \).

Step 5: Calculate the heat energy transferred. Use the formula \( Q = C \cdot \Delta T \), where \( \Delta T \) is the temperature change. The direction of heat transfer is from the hotter substance (oxygen) to the cooler substance (helium).

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Key Concepts

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

Heat Transfer

Heat transfer is the process of thermal energy moving from a hotter object to a cooler one until thermal equilibrium is reached. This can occur through conduction, convection, or radiation. In this scenario, heat will flow from the oxygen (hotter) to the helium (cooler) until both gases reach the same temperature.

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. Different substances have different specific heat capacities, which affect how much heat energy is absorbed or released during temperature changes. This concept is crucial for calculating the heat transfer in the given problem.

Thermal Equilibrium

Thermal equilibrium occurs when two objects in thermal contact no longer exchange heat, meaning they are at the same temperature. In the context of this problem, the heat transfer will continue until the helium and oxygen reach thermal equilibrium, allowing for the calculation of the total heat energy transferred based on their respective masses and specific heat capacities.

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