A Jaguar XK8 convertible has an eight-cylinder engine. At the beginning of its compression stroke, one of the cylinders contains 499 cm ^ 3 of air at atmospheric pressure (1.01\times 10^5 Pa}) and a temperature of 27.0 degrees C. At the end of the stroke, the air has been compressed to a volume of 46.2 cm ^ 3 and the gauge pressure has increased to 2.72\times 10^6 Pa.

Question

A Jaguar XK8 convertible has an eight-cylinder engine. At the beginning of its compression stroke, one of the cylinders contains 499 cm ^ 3 of air at atmospheric pressure (1.01\times 10^5 Pa}) and a temperature of 27.0 degrees C. At the end of the stroke, the air has been compressed to a volume of 46.2 cm ^ 3 and the gauge pressure has increased to 2.72\times 10^6 Pa.

Answer 1

Compute the final temperature.

Answer 2

To analyze this scenario, we can use the ideal gas law equation, which is as follows:

PV = nRT

Where:
P = Pressure
V = Volume
n = Number of moles
R = Ideal gas constant
T = Temperature in Kelvin (K)

To find the number of moles (n), we can rearrange the equation as follows:

n = PV / RT

Given that the initial conditions are:
P1 = 1.01 × 10^5 Pa (atmospheric pressure)
V1 = 499 cm^3
T1 = 27.0 degrees Celsius = 300.15 K

And the final conditions are:
P2 = 2.72 × 10^6 Pa (gauge pressure)
V2 = 46.2 cm^3
T2 = ?

First, we need to convert all the given volumes to m^3 and pressures from Pa to gauge pressure (difference from atmospheric):

V1 = 499 cm^3 = 499 × 10^-6 m^3
P1 = 1.01 × 10^5 Pa (atmospheric pressure)

Next, we need to convert the final gauge pressure (P2) back to absolute pressure:

P2 = 2.72 × 10^6 Pa + 1.01 × 10^5 Pa = 2.83 × 10^6 Pa

Now, we can find the number of moles at the initial conditions:

n = P1V1 / RT1

To find the value of R, we can use the value for the ideal gas constant: R = 8.314 J/(mol·K)

Substituting the values:

n = (1.01 × 10^5 Pa) × (499 × 10^-6 m^3) / [(8.314 J/(mol·K)) × 300.15 K]

Next, we can calculate the temperature at the final conditions using the ideal gas law equation:

T2 = P2V2 / (nR)

Substituting the known values:

T2 = (2.83 × 10^6 Pa) × (46.2 × 10^-6 m^3) / (n × 8.314 J/(mol·K))

Now, we can substitute the calculated value of n into the equation above to find T2.

Finally, we can convert the temperature T2 from Kelvin to degrees Celsius by subtracting 273.15.

Please note that the provided values are rounded, and for more accurate results, you should use the exact values without rounding. Additionally, conversion factors are commonly used in equations, and they are smaller for cm^3 to m^3 and Pa to atm conversion, and larger for K to °C conversion.