Given:
Initial temperature (T1) = 27 C
Initial pressure (P1) = P
Final pressure (P2) = 1/8 of P
Ratio of specific heat capacities (Cp/Cv) = 5/3

To find:
Final temperature (T2)

Solution:

Step 1: Convert initial temperature to Kelvin
Given initial temperature, T1 = 27 C
To convert Celsius to Kelvin, we use the formula:
T(K) = T(C) + 273.15
So, T1(K) = 27 + 273.15 = 300.15 K

Step 2: Using the ideal gas law equation
The ideal gas law equation is given as:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles
R = ideal gas constant
T = temperature

Since we are only interested in the change in temperature, we can rewrite the equation as:
P1V1/T1 = P2V2/T2

Step 3: Analyzing the given information
Given that the pressure is compressed suddenly, we can assume that the volume remains constant (V1 = V2).

Step 4: Applying the given information to the equation
P1/T1 = P2/T2

We know that P2 = 1/8 * P1
Substituting this value in the equation:
P1/T1 = (1/8 * P1)/T2

Step 5: Solving for T2
Let's simplify the equation:
8P1/T1 = P1/T2

Cross-multiplying the equation:
8P1 * T2 = P1 * T1

Dividing both sides of the equation by P1:
8T2 = T1

Step 6: Solving for T2
T2 = T1/8

Step 7: Converting the final temperature to Celsius
To convert Kelvin to Celsius, we use the formula:
T(C) = T(K) - 273.15
So, T2(C) = T2(K) - 273.15 = (T1/8) - 273.15

Final Answer:
The final temperature will be (T1/8) - 273.15 Celsius.

130K