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A nearly flat bicycle tire becomes noticeably warmer after it has been pumped up. Approximate this process as a reversible adiabatic compression. Take the initial pressure and temperature of the air before it is put in the tire to be Pi = 1.00 bar and Ti = 298 K. The final volume of the air in the tire is Vf= 1.50 L and the final pressure is Pf = 5.00 bar. Calculate the final temperature of the air in the tire.
Assume that Cv,m = 5R/2
Answer:
The value is [tex]T_f = 471.978 \ K[/tex]
Explanation:
From the question we are told that
The initial pressure is [tex]P_i = 1.00 \ bar[/tex]
The initial temperature is [tex]T_i = 298 K[/tex]
The final volume is [tex]V_f = 1.50 \ L[/tex]
The final pressure is [tex]P_f = 5.0 \ bar[/tex]
Generally the equation for reversible adiabatic compression is
[tex]P_i V_i^{\gamma} = P_f V_f^{\gamma} [/tex]
Generally from the ideal gas equation we have that
[tex]PV = n RT[/tex]
=> [tex] V = \frac{n RT}{P}[/tex]
So
[tex]P_i *[ \frac{n RT_i}{P_i}]^{\gamma} = P_f [ \frac{n RT_f}{P_f}]^{\gamma}[/tex]
=> [tex]P_i ^{1 - \gamma} * T_i ^{\gamma} = P_f ^{1-\gamma} * T_f^{\gamma}[/tex]
=> [tex]T_f ^{\gamma} = \frac{ P^{1 - \gamma * T_i^{\gamma}}}{ P_f ^{1 - \gamma}}[/tex]
=> [tex]T_f =[ \frac{ P_i ^{1 - \gamma} * T_i^{\gamma}}{P_f^{1 - \gamma}}]^{\frac{1}{\gamma} }[/tex]
=> [tex]T_f = T_i * [\frac{P_i}{P_f} ]^{\frac{1- \gamma}{\gamma}[/tex]
Here [tex]\gamma[/tex] is a constant mathematically represented as
[tex]\gamma = \frac{C_{P,m}}{C_{V, m}}[/tex]
Here
[tex]C_{P,m}} [/tex] is the molar heat capacity at constant pressure
and [tex]C_{V,m}} [/tex] is the molar heat capacity at constant volume given as
[tex]C_{V,m}} = \frac{5R}{2} [/tex]
Generally [tex]C_{P,m}} [/tex] for an ideal gas is mathematically represented as
[tex]C_{P,m}} = R +C_{V,m}} [/tex]
So
[tex]\gamma = \frac{\frac{5R}{2} + R}{ \frac{5R}{2} }[/tex]
=> [tex]\gamma = \frac{7}{5} [/tex]
So
=> [tex]T_f = T_i * [\frac{P_i}{P_f} ]^{\frac{1- \frac{7}{5}}{ \frac{7}{5}}[/tex]
=> [tex]T_f = 298 * [\frac{1}{5} ]^{\frac{1- \frac{7}{5}}{ \frac{7}{5}}[/tex]
=> [tex]T_f = 471.978 \ K[/tex]
The Final temperature of the air in the tire = 471.978k
Given data :
Initial pressure ( [tex]P_{i}[/tex] ) = 1.00 bar
Initial temperature ( [tex]T_{i}[/tex] ) = 298 K
Final volume of air ( [tex]V_{f}[/tex] ) = 1.50 L
Final pressure ( [tex]P_{f}[/tex] ) = 5.00 bar
Assume Cvm = 5r / 2 ( missing data )
Determine the final temperature of the air in the tire
Applying the equation for a reversible Adiabatic compression and general ideal gas equation
[tex]Pi * [ \frac{nRTi}{Pi} ] \alpha = Pf * [ \frac{nRTf}{Pf}] \alpha[/tex] ---- ( 1 )
Resolving equation ( 1 )
[tex]final temperature = Ti * [ \frac{Pi}{Pf}]^{\frac{1-\alpha }{\alpha } }[/tex] ------- ( 2 )
where [tex]\alpha[/tex] ( mathematical constant ) = ( Cpm / Cvm ) --- ( 3 )
Cpm = molar heat capacity at constant pressure = R + Cvm
Cvm = molar heat capacity at constant volume = 5r / 2
∴ Equation ( 3 ) becomes
[tex]\alpha[/tex] = [tex]\frac{7}{5}[/tex]
Final step : Determine the final temperature of the air in the tire
Input values back into equation ( 2 )
[tex]final temperature = 298 + [\frac{1}{5}] ^{\frac{1-\frac{7}{5} }{\frac{7}{5} } }[/tex]
= 471.978 K
Hence we can conclude that the final temperature of the air in the tire is 471.978 K
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