A typical gas cylinder, such as used in teaching and research labs, is around 150 cm tall and has a diameter of 25 cm (with circular cross section). Consider such a cylinder which has been supplied filled with N2 gas at a pressure of 230 bar. What is the mass of the N2 in the cylinder?

To determine the mass of nitrogen gas (\( N_2 \)) in the cylinder, we'll use the following steps and equations:

### Given Information:
- **Height of cylinder (\( h \))** = 150 cm = 1.5 m
- **Diameter of cylinder (\( d \))** = 25 cm = 0.25 m
- **Pressure of \( N_2 \) gas (\( P \))** = 230 bar = 23,000,000 Pa (since 1 bar = 100,000 Pa)
- **Temperature (\( T \))** = Assume standard room temperature, \( T = 298 \, K \) (approximately 25°C)
- **Universal gas constant (\( R \))** = 8.314 J/(mol·K)
- **Molar mass of nitrogen (\( M_{\text{N}_2} \))** = 28 g/mol = 0.028 kg/mol

We will assume that the gas behaves as an ideal gas, so we can use the **ideal gas law**:

\[
PV = nRT
\]

where:
- \( P \) = pressure of the gas
- \( V \) = volume of the cylinder
- \( n \) = number of moles of the gas
- \( R \) = ideal gas constant
- \( T \) = temperature in Kelvin

### Step 1: Calculate the Volume of the Cylinder
The volume of a cylinder is given by:
\[
V = \pi r^2 h
\]
where \( r \) is the radius of the cylinder. Given that the diameter is 0.25 m:
\[
r = \frac{d}{2} = \frac{0.25 \, \text{m}}{2} = 0.125 \, \text{m}
\]

So, the volume is:
\[
V = \pi (0.125 \, \text{m})^2 (1.5 \, \text{m})
\]
\[
V = \pi \times 0.015625 \, \text{m}^2 \times 1.5 \, \text{m} = 0.0733 \, \text{m}^3
\]

### Step 2: Calculate the Number of Moles of \( N_2 \) Gas
Using the ideal gas law:
\[
n = \frac{PV}{RT}
\]
Substituting the known values:
\[
n = \frac{(23,000,000 \, \text{Pa}) (0.0733 \, \text{m}^3)}{(8.314 \, \text{J/(mol·K)}) (298 \, \text{K})}
\]
\[
n \approx \frac{1,686,900}{2,478.572} \approx 681 \, \text{moles}
\]

### Step 3: Calculate the Mass of the \( N_2 \) Gas
The mass of the gas can be calculated using the molar mass:
\[
\text{Mass} = n \times M_{\text{N}_2}
\]
\[
\text{Mass} = 681 \, \text{moles} \times 0.028 \, \text{kg/mol} \approx 19.07 \, \text{kg}
\]

### Conclusion
The mass of the \( N_2 \) gas in the cylinder is approximately **19.07 kg**.

Would you like more details or have any questions?

Here are some related questions to explore:
1. How does changing the temperature affect the mass of gas in the cylinder?
2. What assumptions are made when using the ideal gas law in this scenario?
3. How would the calculation change if the gas did not behave ideally?
4. How does pressure conversion from bar to Pascal work in this context?
5. What are the real-world implications of storing gases under high pressure?

**Tip:** Always ensure that the units are consistent when using the ideal gas law to avoid calculation errors.

Learn how to calculate the mass of nitrogen gas (N2) in a cylinder using the ideal gas law. This problem explores geometry concepts to find the volume of the cylinder and applies the ideal gas law to determine the number of moles and mass. Ideal for undergraduate physics or chemistry students.

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