Question

Determine the unknown quantity, assuming air behaves as an ideal gas. R¯ = 8.314 kJ kmol−1 K −1 . MWair = 28.97 kg kmol−1 . If needed, average the specific heats between the beginning and ending temperatures.1. At I bar and 22 C, determine o and r. 2. At 10 bar and v 0.25 m3 kg 1, determine T and u. 3. At 600 K, determine cp, Cu, k, and v

Answer 1

Answer:

1) o=1.18 kg/m^3; r=286.987 J/(kg K); 2) T=871.12 K, U=375 kJ; 3) cp=438.793 J/(kg K), cv=430.480 J/(kg K), v=1.0.193, k=52.8135

Explanation:

1)

This questions is fully based on the Ideal Gas Law:

$pV=nRT$

As we need to find initial density, we should change temperature to Kelvins:

$T=22+273=295 K$

In addition, we should re-calculate amount of substance n, using mass m and MWair:

$n=m/MWair$

As it is known, density can be calculated, as:

$o=m/V$

Now, we can use all equations in the ideal gas law:

$pV=\frac{m}{MWair} RT\\pMWair=\frac{m}{V} RT\\o=\frac{m}{V} =\frac{pMWair}{RT}=100*10^3*28.97*10^-3/(8.314*295)=1.18 kg/m^{3}$

To calculate specific gas constant for the given scenario (r), we should do the following:

$r=R/MWair=8.314/(28.97*10^{-3})=286.987 J/(kg*K)$

Note, that we converted MWair to kg/mol using (10^(-3)).

2)

To solve this question, we should use again ideal gas law, from where we can directly find temperature of the gas:

$pV=nRT\\pV=\frac{m}{MWair}RT\\ T=\frac{pVMWair}{mR}=\frac{10*10^{5}*28.97*10^{-3}*0.25}{8.314*1}=871.12 K$

Note, that all values were converted to SI units for the calculations (1 bar= 10^5 Pa).

To find internal energy, we can use the following equation:

$U=\frac{3}{2} nRT=\frac{3}{2} \frac{m}{MWair} RT=\frac{3}{2}\frac{1}{28.97*-10^{-3}}*8.314*871.12=375000 J=375 kJ$

3)

To find specific heat, we can use the following ideal gas laws:

$Cv=U/T=\frac{3/2 nRT}{T}=3/2nR=\frac{3}{2}\frac{m}{M} R\\ cv=Cv/m=\frac{3}{2}\frac{R}{M} =430.480 J/(kg K)$

Then, the cp, can be calculated using cv and R:

$cp=cv+R=438.793 J/(kg K)$

The thermal coefficient v represents the ratio between cp and cv:

v=cp/cv=1.0193

Finally, we can find k:

$k=\frac{v}{v-1}= 52.813$