Question

An insulated cylinder of volume \(60\,\text{cm}^3\) is filled with a gas at \(27^\circ\text{C}\) and \(2\) atmospheric pressure. The gas is then compressed making the final volume \(20\,\text{cm}^3\) while allowing the temperature to rise to \(77^\circ\text{C}\). The final pressure is ____________ atmospheric pressure.

Hint:

When both temperature and volume change, always use the combined gas law.

Answer 1

Correct Answer: 7

To solve this problem, we utilize the ideal gas law, which is stated as \(PV=nRT\). Here \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) is temperature in Kelvin. We assume the gas behaves ideally under given conditions. 

First, convert temperatures from Celsius to Kelvin.  
Initial temperature \(T_1\):
\(T_1=27^\circ\text{C}+273.15=300.15\,\text{K}\)
Final temperature \(T_2\):
\(T_2=77^\circ\text{C}+273.15=350.15\,\text{K}\)

The initial and final conditions are related by the equation:
\(\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}\)
Substitute the known values:
\(\frac{2\times60}{300.15}=\frac{P_2\times20}{350.15}\)
Solve for \(P_2\):
\(P_2=\frac{2\times60\times350.15}{300.15\times20}\)
After calculating, 
\(P_2=7\), which is within the expected range (7,7).
Thus, the final pressure is \(7\) atmospheric pressure.