Monatomic Ideal Gas and Internal Energy

Internal Energy in Monatomic Ideal Gas

• Formula for Total Internal Energy:
  • $U = \frac{3}{2} PV = \frac{3}{2} nRT$
  • Internal energy = Total kinetic energy of gas particles
• Changing Internal Energy:
  • Increase pressure, volume, or temperature

Methods to Change Internal Energy

1. Heating It Up:
   • Heat transfer to gas increases particle movement
   • Achieved by placing gas on a flame or hot plate
2. Performing Work on the Gas:
   • Compressing gas with piston increases energy
   • Energy adds due to piston-gas particle interactions

First Law of Thermodynamics

• Formula: $\Delta U = Q + W$
  • $Q$: Heat added
  • $W$: Work done on the gas
• Energy Flow Considerations:
  • Work done on gas: Energy added
  • Work done by gas: Energy leaves system

Work Done by Gas

• Definition: $W = F \times d$
  • Further expressed as $W = P \times \Delta V$ (constant pressure case)
  • Conditions: Pressure must remain constant

Heat Capacity

Definition

• Heat Capacity (C): $C = \frac{Q}{\Delta T}$
• Molar Heat Capacity (C): $C = \frac{Q}{n \Delta T}$

Heat Capacity Types

1. At Constant Volume ($C_V$):

   • Piston is immobile; no work done
   • Formula: $C_V = \frac{\Delta U}{\Delta T}$
   • For monatomic ideal gas, $C_V = \frac{3}{2}nR$
   • Molar heat capacity: $\frac{3}{2}R$

2. At Constant Pressure ($C_P$):

   • Piston moves; pressure constant
   • Formula: $C_P = \frac{Q}{\Delta T} = \frac{\Delta U + P \Delta V}{\Delta T}$
   • For monatomic ideal gas, $C_P = \frac{5}{2}nR$
   • Molar heat capacity: $\frac{5}{2}R$

Relationship Between $C_P$ and $C_V$

• Difference: $C_P - C_V = nR$
• Molar Difference: $C_P - C_V = R$

Summary

• Internal energy can be altered by heating or doing work on the gas.
• The first law of thermodynamics provides a basis for understanding energy changes.
• Heat capacities vary with conditions (constant volume vs. constant pressure).