Hello,............

At 27 C , a gas is suddenly compresse such that its pressure becomes 1 / 8 th of its original pressure. If r = 5 /3 , what is the final temperature ?

Please explain with proper reason ….

Hello! Let's dive into the problem of finding the final temperature of a gas when its pressure is suddenly reduced to one-eighth of its original value. To tackle this, we can use the principles of the ideal gas law and the relationship between pressure, volume, and temperature, particularly focusing on the adiabatic process since the gas is compressed suddenly.

Understanding the Ideal Gas Law

The ideal gas law is expressed as:

PV = nRT

Where:

• P = pressure
• V = volume
• n = number of moles of gas
• R = universal gas constant
• T = temperature in Kelvin

Adiabatic Process and Relationships

In an adiabatic process, where no heat is exchanged with the surroundings, we can use the following relationship for an ideal gas:

P1 * V1^r = P2 * V2^r

Here, r is the heat capacity ratio (given as 5/3 in this case). We also know that:

P1/T1^(r-1) = P2/T2^(r-1)

We can rearrange this equation to find the final temperature T2.

Given Values

Let's summarize the information we have:

• Initial temperature, T1 = 27°C = 300 K (after converting to Kelvin)
• Initial pressure, P1 = P
• Final pressure, P2 = P/8
• Heat capacity ratio, r = 5/3

Calculating the Final Temperature

We can substitute the known values into the rearranged equation:

P1/T1^(r-1) = P2/T2^(r-1)

Substituting the pressures:

P/T1^(5/3 - 1) = (P/8)/T2^(5/3 - 1)

Now, simplifying:

8/T1^(2/3) = T2^(2/3)

Next, we can express T2 in terms of T1:

T2 = 8^(3/2) * T1^(2/3)

Substituting the Values

Now, substituting T1 = 300 K:

T2 = 8^(3/2) * (300)^(2/3)

Calculating 8^(3/2) gives us:

8^(3/2) = 22.6274

Next, we calculate (300)^(2/3):

(300)^(2/3) ≈ 173.205

Now, multiplying these two results:

T2 ≈ 22.6274 * 173.205 ≈ 3915.4 K

Final Result

Thus, the final temperature of the gas after the pressure is reduced to one-eighth of its original value is approximately 3915.4 K. This significant increase in temperature illustrates how compressing a gas can lead to a rise in temperature, especially in an adiabatic process where no heat is exchanged with the environment.