A cylinder contains the helium gas with a volume of 0.1 m3 at a pressure of 220 kPa.
The gas is compressed to a volume of 0.02 m3. The process is adiabatic and the
specific heat ratio of the gas is 1.66. Determine
(a) the pressure in the cylinder after compression;
(b) work done to reduce the volume of the gas; and
(c) percentage change of the temperature.
(a) Use the relation P V^1.66 = constant
(b) Work = Integral of P*dV. Express P as a function of V to do the integration
(c) Use the relation PV/T = constant to get the new T.
OR use the isentropic compression relation
(T2/T1) = (V1/V2)^(g-1)
where g = 1.66
You should get the same answer either way.
To solve this problem, we can use the adiabatic compression formula:
\( P_1 \cdot V_1^{\gamma} = P_2 \cdot V_2^{\gamma} \)
where:
- \( P_1 \) and \( P_2 \) are the initial and final pressures respectively
- \( V_1 \) and \( V_2 \) are the initial and final volumes respectively
- \( \gamma \) is the specific heat ratio
Given:
\( V_1 = 0.1 \, \text{m}^3 \)
\( P_1 = 220 \, \text{kPa} \)
\( V_2 = 0.02 \, \text{m}^3 \)
\( \gamma = 1.66 \)
(a) To find the pressure in the cylinder after compression, we can rearrange the formula as follows:
\( P_2 = P_1 \cdot \left(\frac{V_1}{V_2}\right)^{\gamma} \)
Substituting the given values, we get:
\( P_2 = 220 \, \text{kPa} \cdot \left(\frac{0.1 \, \text{m}^3}{0.02 \, \text{m}^3}\right)^{1.66} \)
Using a calculator, we can find the value of \( P_2 \).
(b) To find the work done to reduce the volume of the gas, we can use the formula:
\( W = \frac{{P_2 \cdot V_2 - P_1 \cdot V_1}}{{\gamma - 1}} \)
Substituting the given values, we get:
\( W = \frac{{P_2 \cdot 0.02 \, \text{m}^3 - 220 \, \text{kPa} \cdot 0.1 \, \text{m}^3}}{{1.66 - 1}} \)
Again, using a calculator, we can find the value of \( W \).
(c) The percentage change of temperature can be calculated using the formula:
\( \frac{{T_2 - T_1}}{{T_1}} \times 100 \)
where:
- \( T_1 \) and \( T_2 \) are the initial and final temperatures respectively
Since the process is adiabatic, there is no heat exchange, which means the temperature remains constant. Therefore, the percentage change of temperature is 0%.
Using these formulas and calculations, you can find the answers to parts (a), (b), and (c) of the problem.