A cylinder in a car engine takes Vi = 4.50 10-2 m3 of air into the chamber at 30° C and at atmospheric pressure. The piston then compresses the air to 0.112 times of the original volume (0.112 Vi) and to 24.0 times the original pressure (24.0 Pi). What is the new temperature of the air?
T1 =273+30 =303 K
p1•V1/T1 = p2•V2/T2
T2 = T1• p2•V2/ p1•V1=
=303•24•p1•0.112V1/ p1•V1=814.5 K
Well, you've come to the right place if you're looking for some hot air! Let's dive into this cylinder conundrum.
Now, we know that the volume and pressure of the air have changed, but we're missing one crucial component - the number of moles of air in the cylinder. Without this information, we can't determine the new temperature. So, we'll have to head back to the drawing board (or engineering lab) on this one. Keep up the good work, and remember, there's always a solution hiding somewhere - just like the punchline to a good joke!
To find the new temperature of the air, we can use the ideal gas law equation:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles
R = ideal gas constant
T = temperature
Since the number of moles of air remains constant, we can rewrite the equation as:
P1V1/T1 = P2V2/T2
Where:
P1 = initial pressure
V1 = initial volume
T1 = initial temperature
P2 = final pressure
V2 = final volume
T2 = final temperature
From the given information, we know:
V1 = Vi = 4.50 x 10^-2 m^3
P1 = atmospheric pressure
V2 = 0.112 Vi
P2 = 24.0 Pi
To begin, let's find the value of Pi. Atmospheric pressure can vary, but a commonly used value is around 1.013 x 10^5 Pa. Therefore, we have:
Pi = 1.013 x 10^5 Pa
Next, calculate the final pressure P2:
P2 = 24.0 Pi
= 24.0 x 1.013 x 10^5 Pa
= 2.4312 x 10^6 Pa
Now, we can rearrange the equation to solve for the final temperature T2:
P1V1/T1 = P2V2/T2
T2 = P2V2(T1)/(P1V1)
Substituting in the known values:
T2 = (2.4312 x 10^6 Pa) * (0.112 Vi) * (T1) / (atmospheric pressure * Vi)
Simplifying further:
T2 = (2.4312 x 10^6) * (0.112 T1) / (atmospheric pressure)
Now, substitute the known value for atmospheric pressure:
T2 = (2.4312 x 10^6) * (0.112 T1) / (1.013 x 10^5 Pa)
Calculating:
T2 = 269.18 T1 / 1.013
Therefore, the new temperature of the air is 269.18 times the initial temperature (T1).
To find the new temperature of the air, we can use the ideal gas law equation: PV = nRT.
First, let's calculate the initial conditions:
Initial volume (Vi) = 4.50 × 10^(-2) m^3
Initial temperature (Ti) = 30°C = 30 + 273.15 = 303.15 K
Initial pressure (Pi) = atmospheric pressure
Next, let's calculate the final conditions:
Final volume (Vf) = 0.112 × Vi
Final pressure (Pf) = 24.0 × Pi
Since the amount of gas (moles, n) remains constant, we can rewrite the ideal gas law equation as:
(Pi × Vi) / Ti = (Pf × Vf) / Tf
Now, we can rearrange the equation to solve for the final temperature (Tf):
Tf = (Pf × Vf × Ti) / (Pi × Vi)
Substituting the given values:
Tf = (24.0 × Pi × 0.112 × Vi × Ti) / (Pi × Vi)
Canceling out Pi and Vi:
Tf = 24.0 × 0.112 × Ti
Substituting the value of Ti = 303.15 K into the equation:
Tf = 24.0 × 0.112 × 303.15 K
Tf ≈ 814.23 K
So, the new temperature of the air is approximately 814.23 K.