Chapter 19: Problem 22
Three moles of an ideal monatomic gas expands at a constant pressure of 2.50 atm; the volume of the gas changes from 3.20 \(\times\) 10\(^{-2}\) m\(^3\) to 4.50 \(\times\) 10\(^{-2}\) m\(^3\). Calculate (a) the initial and final temperatures of the gas; (b) the amount of work the gas does in expanding; (c) the amount of heat added to the gas; (d) the change in internal energy of the gas.
Short Answer
Step by step solution
Understand Given Values
Use Ideal Gas Law for Initial Temperature
Use Ideal Gas Law for Final Temperature
Calculate Work Done by Gas
Calculate Heat Added to Gas
Find Change in Internal Energy
Final Summary
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Monatomic Gas
These gases are significant in thermodynamics studies due to their simplicity. The absence of bonds between atoms means there are no molecular vibrations or rotations contributing to internal energy. Only the kinetic energy of the gas particles needs to be considered.
- Monatomic gases have specific heat capacities that depend only on translation movements.
- At constant pressure, the molar heat capacity \(C_p\) for a monatomic gas is \(\frac{5}{2}R\).
- At constant volume, \(C_v\) is \(\frac{3}{2}R\).
This simplicity makes them an excellent model for exploring gas behaviors and thermodynamic processes.
Work Done by Gas
\[W = P(V_f - V_i)\]
where:
- \(P\) is the constant pressure
- \(V_f\) and \(V_i\) are the final and initial volumes, respectively.
This work is a form of energy transfer from the gas to its surroundings. Understanding this helps in analyzing the efficiency of engines and other thermodynamic systems.
Internal Energy Change
\[\Delta U = nC_v\Delta T\]
where:
- \(\Delta T = T_f - T_i\) is the change in temperature
- \(C_v = \frac{3}{2}R\) is the specific heat at constant volume
This concept is vital because the sum of energy added as heat and energy done as work equals the change in the internal energy, embodying the first law of thermodynamics, \(\Delta U = Q - W\).
Constant Pressure Expansion
In constant pressure processes, the amount of heat \(Q\) added to the system can be calculated from:
\[Q = nC_p\Delta T\]
where:
- \(C_p = \frac{5}{2}R\) is the specific heat at constant pressure
- \(n\) is the number of moles
- \(\Delta T\) is the temperature change