To solve the problem, we need to use the principles of adiabatic processes in thermodynamics for an ideal gas. An adiabatic process is one in which no heat is transferred to or from the gas, which means that all the work done on the gas is converted into a change in internal energy, leading to a change in temperature.
Given:
- The ratio of heat capacities, [tex]\( \gamma = \frac{C_p}{C_v} = \frac{5}{3} \)[/tex]
- Initial temperature, [tex]\( T_1 = 72^\circ C \)[/tex]
- The volume is expanded to eight times its original volume, which means [tex]\( \frac{V_2}{V_1} = 8 \)[/tex]
We are to find the rise in temperature of the gas. First, let's convert the initial temperature to Kelvin since temperature in thermodynamic equations should be in absolute scale:
[tex]\[ T_1 (K) = 72^\circ C + 273.15 = 345.15 \, K \][/tex]
For an adiabatic process, the relation between initial and final temperatures and volumes is given by:
[tex]\[ \frac{T_2}{T_1} = \left( \frac{V_1}{V_2} \right)^{\gamma - 1} \][/tex]
Since [tex]\( V_2 = 8V_1 \)[/tex], we can write:
[tex]\[ \frac{T_2}{T_1} = \left( \frac{V_1}{8V_1} \right)^{\gamma - 1} = \left( \frac{1}{8} \right)^{\frac{5}{3} - 1} = \left( \frac{1}{8} \right)^{\frac{2}{3}} \][/tex]
Let’s denote this exponentiation for clarity:
[tex]\[ \left( \frac{1}{8} \right)^{\frac{2}{3}} \][/tex]
Using this relation, we find the final temperature [tex]\( T_2 \)[/tex]:
[tex]\[ T_2 = T_1 \times \left( \frac{1}{8} \right)^{\frac{2}{3}} \][/tex]
Now, substituting [tex]\( T_1 = 345.15 \, K \)[/tex]:
[tex]\[ T_2 = 345.15 \times \left( \frac{1}{8} \right)^{\frac{2}{3}} \][/tex]
After computation, we get:
[tex]\[ T_2 \approx 86.29 \, K \][/tex]
The rise in temperature can be found by:
[tex]\[ \Delta T = T_2 - T_1 = 86.29 \, K - 345.15 \, K = -258.8625 \, K \][/tex]
Since this is a decrease in temperature (as expected in adiabatic expansion where the gas cools down), the approximate change in temperature is a decrease of:
[tex]\[ -258.86 \, K \][/tex]
None of the provided options (a. [tex]$86 k$[/tex], b. [tex]$186 k$[/tex], c. [tex]$299 k$[/tex], d. [tex]$273 k$[/tex]) accurately describe a temperature change given that they are positive values and our calculated value is negative. Thus, the closest general understanding, if we were to consider the magnitude of temperature change as a rise in context, would be approximately [tex]$258.86 \, K$[/tex], which does not match exactly any of the given options reflecting a rise.
