Question

A cylinder contains 0.100 mol of an ideal monatomic gas. Initially the gas is at 1.00×1051.00\$$times10^5 Pa $$and occupies a volume of 2.50×10−32.50\$$times10^{-3} m3. If $$the gas is allowed to expand to twice the initial volume, find the final temperature (in kelvins) and pressure of the gas if the expansion is (i) isothermal; (ii) isobaric; (iii) adiabatic.

Accepted Answer

Step 1: Begin by identifying the given values and the relationships between pressure, volume, and temperature for an ideal gas. The initial pressure (P₁) is 1.00 × 10⁵ Pa, the initial volume (V₁) is 2.50 × 10⁻³ m³, and the initial number of moles (n) is 0.100 mol. Use the ideal gas law, $$ PV = nRT $$, to calculate the initial temperature $$ T₁ . $$Rearrange the formula to $$ T₁ = \frac{P₁V₁}{nR} $$, where $$ R  is $$the universal gas constant (8.314 J/(mol·K)). Step 2: For part (i), the isothermal expansion means the temperature remains constant. Since $$ T  is $$constant, the ideal gas law simplifies to $$ P₁V₁ = P₂V₂ . $$The final volume $$ V₂  is $$twice the initial volume, so $$ V₂ = 2V₁ . $$Rearrange the equation to solve for the final pressure $$ P₂ $$: $$ P₂ = \frac{P₁V₁}{V₂} . $$Substitute the known values to find $$ P₂ . $$The final temperature $$ T $$ remains equal to $$ T₁ . $$Step 3: For part (ii), the isobaric expansion means the pressure remains constant. Since $$ P  is $$constant, the ideal gas law simplifies to $$ \frac{V₁}{T₁} = \frac{V₂}{T₂} . $$Rearrange the equation to solve for the final temperature $$ T₂ $$: $$ T₂ = \frac{V₂T₁}{V₁} . $$Substitute the known values to find $$ T₂ . $$The final pressure $$ P $$ remains equal to $$ P₁ . $$Step 4: For part (iii), the adiabatic expansion means no heat is exchanged with the surroundings. Use the adiabatic condition for an ideal monatomic gas: $$ P₁V₁^\gamma = P₂V₂^\gamma $$, where $$ \gamma = \frac{5}{3} $$ for a monatomic gas. Rearrange the equation to solve for $$ P₂ $$: $$ P₂ = P₁ \left( \frac{V₁}{V₂} \right)^\gamma . $$Substitute the known values to find $$ P₂ . $$Then use the ideal gas law $$ T₂ = \frac{P₂V₂}{nR}  to $$find the final temperature $$ T₂ . $$Step 5: Summarize the results for each case: (i) Isothermal expansion keeps temperature constant and uses $$ P₁V₁ = P₂V₂  to $$find $$ P₂ . (ii) $$Isobaric expansion keeps pressure constant and uses $$ \frac{V₁}{T₁} = \frac{V₂}{T₂}  to $$find $$ T₂ . ($$iii) Adiabatic expansion uses $$ P₁V₁^\gamma = P₂V₂^\gamma  to $$find $$ P₂ $$, and $$ T₂ = \frac{P₂V₂}{nR}  to $$find $$ T₂ .$$

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Key Concepts

Ideal Gas Law

Types of Thermodynamic Processes

Monatomic Ideal Gas Properties