To find the specific heat capacity at constant volume (CV) for the gas under the given conditions, we can use the relationships between the specific heat capacities, the adiabatic process, and the ideal gas law. Let's break this down step by step.
Understanding the Problem
We know that at standard temperature and pressure (STP), one mole of gas has a mass of 273 grams. The gas expands adiabatically, meaning there is no heat exchange with the surroundings. The volume of the gas is tripled, and we have the following constants:
- Specific heat at constant pressure (Cp) = 1008 J/kg·K
- Ratio of specific heats (γ) = 1.41
- Constant in T-V relationship = 0.025
Key Relationships
In thermodynamics, the relationship between the specific heat capacities at constant pressure (Cp) and constant volume (CV) is given by:
γ = Cp / CV
From this equation, we can rearrange it to find CV:
CV = Cp / γ
Calculating CV
Now, substituting the known values into the equation:
CV = 1008 J/kg·K / 1.41
Calculating this gives:
CV ≈ 714.18 J/kg·K
Final Thoughts
The specific heat capacity at constant volume (CV) for the gas, after it has expanded adiabatically and tripled its volume, is approximately 714.18 J/kg·K. This value indicates how much heat energy is required to raise the temperature of the gas by one degree Kelvin while keeping the volume constant.
Why This Matters
Understanding specific heat capacities is crucial in thermodynamics as it helps us predict how gases behave under different conditions. In practical applications, this knowledge is essential for designing engines, refrigerators, and other systems that rely on gas behavior.
