Compressing gases requires work and the resulting energy is usually converted to heat; if this heat does not escape, the gas's temperature will rise. This effect is used in diesel engines: The compressed air gets so hot that when the fuel is injected, it ignites without any spark plugs.

As an example, consider a cylinder in a diesel engine in which air is compressed to one twentieth of its original volume while the pressure rises from 1 atm to 40 atm (absolute not gauge). Note that because the air heats up while being compressed, its pressure rises more than twenty-fold.
If the air is taken into the cylinder at 11 degrees Celsius, how hot does it get after being compressed? Answer in degrees Celsius

To determine how hot the air gets after being compressed, we can use the ideal gas law, which states:

PV = nRT

where:
P is the pressure of the gas
V is the volume of the gas
n is the number of moles of the gas
R is the ideal gas constant
T is the temperature of the gas

In this case, we know that the volume has decreased to 1/20th of its original volume, so the final volume (Vf) is 1/20th of the initial volume (Vi). We also know that the pressure has increased to 40 atm. From this information, we can set up the following equation:

(Vi)(Pi) = (Vf)(Pf)

To find the final temperature (Tf), we rearrange the ideal gas law equation to solve for T:

Tf = (Vf)(Pf)/(nR)

First, we need to calculate the initial volume (Vi) and final volume (Vf):

Vi = Vf * 20
Vi = 1/20 * Vi = 0.05 * Vi

Now let's consider the pressure. Since the pressure rises more than twenty-fold, we can assume that the pressure follows a linear relationship with the volume. So the final pressure (Pf) should be 40 times the initial pressure (Pi):

Pf = 40 * Pi

Additionally, we know the air is taken into the cylinder at 11 degrees Celsius, which we need to convert to Kelvin before using in the equation:

T_initial = 11 + 273.15 = 284.15 K

Next, we need to determine the value of the ideal gas constant (R). The ideal gas constant is typically expressed in different units, but we can use the value 0.0821 L.atm/mol.K.

Finally, we can plug in all the values into the equation:

T_final = (Vf)(Pf)/(nR)
T_final = (0.05 * Vi) * (40 * Pi) / (n * R)

Unfortunately, we do not have information about the number of moles of gas (n). Without this information, it is not possible to determine the exact temperature.