Question

A 100 cm³ box contains helium at a pressure of 2.0 atm and a temperature of 100℃. It is placed in thermal contact with a 200 cm³ box containing argon at a pressure of 4.0 atm and a temperature of 400℃. How much heat energy is transferred, and in which direction?

Accepted Answer

Step 1: Begin by identifying the key variables for each gas. For helium, the initial pressure (P₁) is 2.0 atm, the initial temperature (T₁) is 100℃ (convert to Kelvin: T₁ = 100 + 273 = 373 K), and the volume (V₁) is 100 cm³. For argon, the initial pressure (P₂) is 4.0 atm, the initial temperature (T₂) is 400℃ (convert to Kelvin: T₂ = 400 + 273 = 673 K), and the volume (V₂) is 200 cm³. Step 2: Use the ideal gas law, $$ PV = nRT $$, to calculate the number of moles (n) for each gas. For helium: $$ n₁ = \frac{P₁ V₁}{RT₁} $$, where R is the universal gas constant (8.314 J/(mol·K)). For argon: $$ n₂ = \frac{P₂ V₂}{RT₂} . $$Substitute the given values to find $$ n₁ $$ and $$ n₂ . $$Step 3: Determine the final equilibrium temperature (Tₓ) after thermal contact. Since heat transfer occurs until thermal equilibrium is reached, use the principle of conservation of energy: $$ Q_{helium} + Q_{argon} = 0 . $$The heat transferred for each gas is given by $$ Q = nC_v \Delta T $$, where $$ C_v  is $$the molar specific heat at constant volume. For monatomic gases like helium and argon, $$ C_v = \frac{3}{2}R . $$Set up the equation $$ n₁C_v(Tₓ - T₁) + n₂C_v(Tₓ - T₂) = 0 $$ and solve for $$ Tₓ . $$Step 4: Calculate the heat energy transferred for each gas using $$ Q = nC_v \Delta T . $$For helium: $$ Q_{helium} = n₁C_v(Tₓ - T₁) . $$For argon: $$ Q_{argon} = n₂C_v(Tₓ - T₂) . $$Substitute the values of $$ n₁ $$, $$ n₂ $$, $$ C_v $$, and $$ Tₓ  to $$find the magnitude of heat transferred for each gas. Step 5: Determine the direction of heat transfer. If $$ Q_{helium} \u003e 0 $$, heat is absorbed by helium, and if $$ Q_{helium} \u003c 0 $$, heat is released by helium. Similarly, analyze $$ Q_{argon} . $$The direction of heat transfer will be from the hotter gas (argon) to the cooler gas (helium) until equilibrium is reached.

A 100 cm³ box contains helium at a pressure of 2.0 atm and a temperature of 100℃. It is placed in thermal contact with a 200 cm³ box containing argon at a pressure of 4.0 atm and a temperature of 400℃. How much heat energy is transferred, and in which direction?

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Key Concepts

Thermal Energy Transfer

Ideal Gas Law

Specific Heat Capacity