A rigid tank containing 0.5 kg of an ideal gas at 100 kPa and 300 K is to be heated to 400 K. The amount of heat transfer required in this process is ___. Assuming that the ideal g... A rigid tank containing 0.5 kg of an ideal gas at 100 kPa and 300 K is to be heated to 400 K. The amount of heat transfer required in this process is ___. Assuming that the ideal gas has Cv = 1.6 kJ/kgK and the heat absorbed by the rigid tank is negligible.
Understand the Problem
The question is asking for the amount of heat transfer required to heat an ideal gas in a rigid tank from 300 K to 400 K. It provides the initial conditions, mass, and specific heat capacity, which must be used to calculate the heat transfer needed in this process.
Answer
$Q = 80 \, \text{kJ}$
Answer for screen readers
The amount of heat transfer required is $Q = 80 , \text{kJ}$.
Steps to Solve
- Determine the mass and specific heat capacity
The mass ($m$) of the gas is given as 0.5 kg. The specific heat capacity at constant volume ($C_v$) is provided as 1.6 kJ/kg·K.
- Calculate the change in temperature
The initial temperature ($T_i$) is 300 K, and the final temperature ($T_f$) is 400 K.
To find the change in temperature ($\Delta T$): $$ \Delta T = T_f - T_i = 400 , \text{K} - 300 , \text{K} = 100 , \text{K} $$
- Use the heat transfer formula
For an ideal gas in a rigid tank, the heat transfer ($Q$) can be calculated using the formula: $$ Q = m \cdot C_v \cdot \Delta T $$
Substituting the values: $$ Q = 0.5 , \text{kg} \cdot 1.6 , \text{kJ/kg·K} \cdot 100 , \text{K} $$
- Calculate the heat transfer
Perform the multiplication: $$ Q = 0.5 \cdot 1.6 \cdot 100 = 80 , \text{kJ} $$
The amount of heat transfer required is $Q = 80 , \text{kJ}$.
More Information
In processes involving an ideal gas in a rigid tank, the heat transfer is directly related to the mass, specific heat capacity, and the temperature change. This type of problem helps illustrate the principles of thermodynamics and heat transfer.
Tips
- Ignoring the rigid tank condition: Since the tank is rigid, volume doesn't change, and the formula used must account for constant volume conditions.
- Incorrectly calculating $\Delta T$: Ensure to subtract the initial temperature from the final temperature correctly.
- Misapplication of units: Make sure to keep an eye on units (e.g., kJ, kg, K) to avoid mistakes in calculations.
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