A steel cylinder contains an ideal gas at 27°C. The gauge pressure is 140kPa. If the temperature of the container increases to 79°C, what is the new gauge pressure?
As we have an ideal gas, we can assume that (P/T)initial = (P/T)final
First, we must convert temperatures in Celsius to Kelvin.
27ºC = (27 + 273.15)K = 300.15K
79ºC = (79+273.15)K = 352.15K
Let x represent (T)final
300.15 K ÷ 140,000 Pa = 352.15K ÷ x
==> x = 164,255 Pa
To find the new gauge pressure, we can use the ideal gas law equation:
P1 * V1 / T1 = P2 * V2 / T2
where P1 is the initial pressure, V1 is the initial volume, T1 is the initial temperature, P2 is the final pressure, V2 is the final volume, and T2 is the final temperature.
In this case, the volume of the cylinder remains constant, so we can assume V1 = V2. Also, the question asks for the gauge pressure change, so we treat the initial gauge pressure (P1) and the final gauge pressure (P2) as absolute pressures by adding atmospheric pressure (which we assume to be 101.3 kPa) to them.
Now let's calculate the new gauge pressure:
Step 1: Convert initial pressure to absolute pressure:
Initial absolute pressure (P1_abs) = P1 + atmospheric pressure
= 140 kPa + 101.3 kPa
= 241.3 kPa
Step 2: Convert temperatures to Kelvin scale:
Initial temperature (T1) = 27°C + 273.15
= 300.15 K
Final temperature (T2) = 79°C + 273.15
= 352.15 K
Step 3: Use the ideal gas law equation to find the final absolute pressure (P2_abs):
P1_abs * V1 / T1 = P2_abs * V2 / T2
P2_abs = P1_abs * T2 / T1
= 241.3 kPa * 352.15 K / 300.15 K
Step 4: Calculate the final gauge pressure (P2) by subtracting atmospheric pressure:
Final gauge pressure (P2) = P2_abs - atmospheric pressure
Now let's substitute the values and calculate the new gauge pressure:
P2_abs = 241.3 kPa * 352.15 K / 300.15 K
= 282.45 kPa
P2 = P2_abs - atmospheric pressure
= 282.45 kPa - 101.3 kPa
= 181.15 kPa
Therefore, the new gauge pressure is approximately 181.15 kPa.