$$ 4.3 \quad \begin{array}{l}0,61 \mathrm{~kg} ext { of nitrogen at an absolute pressure of } 100 \mathrm{kPa} ext { and with a volume of } \ 521 ext { litres has a temperature of } 27^{\circ} \mathrm{C}\end{array} $$ Calculate the gas constant for nitrogen. [10] TOTAL: 60

Question

$$ 4.3 \quad \begin{array}{l}0,61 \mathrm{~kg} 	ext { of nitrogen at an absolute pressure of } 100 \mathrm{kPa} 	ext { and with a volume of } \ 521 	ext { litres has a temperature of } 27^{\circ} \mathrm{C}\end{array} $$ Calculate the gas constant for nitrogen. [10] TOTAL: 60

UpStudy AI · Accepted Answer

1. Write down the ideal gas law in the form that relates pressure, volume, mass, the specific gas constant, and temperature:
   $$
   pV = mRT
   $$
   where  
   $$ p $$ is the pressure,  
   $$ V $$ is the volume,  
   $$ m $$ is the mass,  
   $$ R $$ is the specific gas constant for nitrogen, and  
   $$ T $$ is the absolute temperature.

2. Rearranging for $$ R $$ gives:
   $$
   R = \frac{pV}{mT}
   $$

3. Convert all given quantities to SI units:
   - Pressure:  
     $$ p = 100\,\text{kPa} = 100\,000\,\text{Pa} $$
   - Volume:  
     $$ V = 521\,\text{L} = \frac{521}{1000}\,\text{m}^3 = 0.521\,\text{m}^3 $$
   - Temperature:  
     $$ T = 27^\circ\text{C} = 27 + 273.15 = 300.15\,\text{K} $$
   - Mass is already in kilograms:  
     $$ m = 0.61\,\text{kg} $$

4. Substitute the converted values into the formula for $$ R $$:
   $$
   R = \frac{100\,000 \times 0.521}{0.61 \times 300.15}
   $$

5. Calculate the numerator:
   $$
   100\,000 \times 0.521 = 52\,100\,\text{Pa}\cdot\text{m}^3
   $$

6. Calculate the denominator:
   $$
   0.61 \times 300.15 \approx 183.0915\,\text{kg}\cdot\text{K}
   $$

7. Now, compute $$ R $$:
   $$
   R \approx \frac{52\,100}{183.0915} \approx 284.6\,\text{J}\,\text{kg}^{-1}\,\text{K}^{-1}
   $$

Thus, the gas constant for nitrogen is approximately  
$$
R \approx 285\,\text{J}\,\text{kg}^{-1}\,\text{K}^{-1}.
$$
The gas constant for nitrogen is approximately $$ 285\,\text{J}\,\text{kg}^{-1}\,\text{K}^{-1} $$.

Calculate the gas constant for nitrogen.
[10]
TOTAL:
60

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