Question

Air in a diesel engine cylinder is quickly compressed from an initial temperature of \(20.0^{\circ} \mathrm{C}\), an initial pressure of \(1.00 \mathrm{~atm}\), and an initial volume of \(600 . \mathrm{cm}^{3}\) to a final volume of \(45.0 \mathrm{~cm}^{3}\). Assuming the air to be an ideal diatomic gas, find the final temperature and pressure.

Short answer

Answer: The final temperature is approximately 1354.2 K and the final pressure is approximately 53.6 atm after the adiabatic compression of the air in the diesel engine cylinder.

Step-by-step solution

Write down the initial conditions

We are given the following initial conditions: - Initial temperature \(T_1 = 20.0^{\circ}\mathrm{C} = 293.15\mathrm{K}\) (converted to Kelvin) - Initial pressure \(P_1 = 1.00\mathrm{~atm}\) - Initial volume \(V_1 = 600\mathrm{~cm}^3\) We also have the final volume: - Final volume \(V_2 = 45.0\mathrm{~cm}^3\) In addition, we know that air is an ideal diatomic gas.

Find the final temperature

The compression is adiabatic, meaning there is no heat exchange between the gas and the environment. The relationship between the initial and final conditions for an adiabatic process in an ideal diatomic gas is given by: $$ T_1V_1^{\gamma-1} = T_2 V_2^{\gamma-1} $$ where \(\gamma\) is the adiabatic index for an ideal diatomic gas and is approximately equal to 1.4. We can rearrange the equation and solve for the final temperature, \(T_2\): $$ T_2 = T_1 \left(\frac{V_1}{V_2}\right)^{\gamma - 1} $$ Plugging in the given values: $$ T_2 = 293.15\mathrm{K} \times \left(\frac{600\mathrm{~cm}^3}{45.0\mathrm{~cm}^3}\right)^{1.4 - 1} $$ Calculating the final temperature: $$ T_2 \approx 1354.2\mathrm{K} $$

Find the final pressure

Now that we have the final temperature, we can find the final pressure using the ideal gas law: $$ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} $$ Rearranging the equation to solve for \(P_2\): $$ P_2 = \frac{P_1 V_1 T_2}{T_1 V_2} $$ Plugging in the given values: $$ P_2 = \frac{1.00\mathrm{~atm} \times 600\mathrm{~cm}^3 \times 1354.2\mathrm{K}}{293.15\mathrm{K} \times 45.0\mathrm{~cm}^3} $$ Calculating the final pressure: $$ P_2 \approx 53.6\mathrm{~atm} $$

State the final results

After the adiabatic compression of the air in the diesel engine cylinder, the final temperature and pressure are approximately: - Final temperature: \(T_2 \approx 1354.2\mathrm{K}\) - Final pressure: \(P_2 \approx 53.6\mathrm{~atm}\)

Key concepts

Ideal Diatomic Gas

To understand what an ideal diatomic gas is, we need to break down the phrase into two components: 'ideal gas' and 'diatomic'. When we refer to an 'ideal gas', it's a hypothetical gas composed of many randomly moving point particles that interact only through elastic collisions. This means that the particles don't have any volume (though the gas does) and there's no intermolecular forces between them, except during collisions. The behavior of ideal gases can be described by the ideal gas law, which we'll delve into shortly. Now, 'diatomic' means that each molecule of the gas is made up of two atoms. For instance, the oxygen we breathe, O2, is a diatomic molecule.

In the exercise provided, air (which is a mixture of gases including diatomic species like oxygen and nitrogen) is treated as an ideal diatomic gas. This simplification allows us to use the ideal gas law and other thermodynamic relationships to predict how the gas will behave under different conditions, such as during adiabatic compression in a diesel engine cylinder.

Ideal Gas Law

The ideal gas law is a fundamental equation in thermodynamics and physical chemistry that describes the relationships between pressure (P), volume (V), temperature (T), and the number of moles of gas (n), using the gas constant (R). It's mathematically depicted as

\[ PV = nRT \]

This law allows us to relate the physical properties of an ideal gas in a closed system where the number of particles (n) remains constant. In our exercise, we assume air behaves as an ideal gas, and we use this law to find the final pressure after the gas is adiabatically compressed. Its adaptability to various problems, including those in which the gas undergoes changes in temperature, pressure, or volume, makes it an invaluable tool in understanding how gases will react under different conditions without needing to account for the complexities of real gas behaviors.

Thermodynamics

Thermodynamics is the branch of physics concerned with heat and temperature and their relation to energy and work. It involves the study of the movement of heat through physical systems and the resultant work done and energy changes. There are several laws of thermodynamics, but this exercise focuses on the first law, which asserts that energy cannot be created or destroyed, only transformed.

An adiabatic process, as mentioned in the exercise, is a thermodynamic scenario where no heat is lost or gained by the system from its surroundings. In other words, all the work done on or by the system results solely in changes in internal energy, and not from heat exchange. This is crucial in understanding how the temperature and pressure of the air in the diesel engine's cylinder change: the work done by compressing the gas increases its internal energy, thus increasing the temperature and pressure of the gas even though no heat is added to the system.