To determine the length of the gas column in the cylindrical vessel after evacuating it, we need to apply the principles of gas laws and hydrostatic pressure. Let's break down the problem step by step.
Understanding the System
We have a cylindrical vessel with an ideal gas and a piston. The key parameters are:
- Cross-sectional area of the piston, A = 10 cm² = 10 × 10-4 m²
- Weight of the piston, W = 1 kg
- Atmospheric pressure, Patm = 100 kPa = 100,000 Pa
Calculating the Force Exerted by the Piston
The force exerted by the piston due to its weight can be calculated using the formula:
F = m × g
Where:
- m = mass of the piston = 1 kg
- g = acceleration due to gravity ≈ 9.81 m/s²
Thus, the force is:
F = 1 kg × 9.81 m/s² = 9.81 N
Finding the Pressure Exerted by the Piston
The pressure exerted by the piston on the gas can be calculated using the area of the piston:
Ppiston = F / A
Substituting the values:
Ppiston = 9.81 N / (10 × 10-4 m²) = 981 Pa
Calculating the Total Pressure on the Gas
The total pressure acting on the gas is the sum of the atmospheric pressure and the pressure due to the piston:
Ptotal = Patm + Ppiston
Substituting the values:
Ptotal = 100,000 Pa + 981 Pa = 100,981 Pa
Using the Ideal Gas Law
Since the temperature remains constant, we can use the ideal gas law in the form:
P × V = n × R × T
Where:
- P = total pressure
- V = volume of the gas
- n = number of moles of gas (constant)
- R = ideal gas constant
- T = temperature (constant)
Relating Volume to Length
The volume of the gas in the cylindrical vessel can be expressed as:
V = A × L
Where L is the length of the gas column. Rearranging the ideal gas law gives:
L = (n × R × T) / P
Calculating the Length of the Gas Column
Assuming we have 1 mole of gas (for simplicity), and using R = 8.314 J/(mol·K) and T = 273 K (standard conditions), we can substitute these values into the equation:
L = (1 mol × 8.314 J/(mol·K) × 273 K) / 100,981 Pa
Calculating this gives:
L ≈ 0.0224 m
Final Calculation for Length
To convert this to kilometers:
L ≈ 0.0224 m = 0.0000224 km
However, if we consider the pressure difference and the fact that the gas expands significantly when evacuated, the length can be much larger. The calculated length of 2.2 km seems to arise from a misunderstanding of the conditions or assumptions made in the problem. The actual physical constraints and the behavior of the gas under these conditions would need to be revisited to clarify the discrepancy.
In conclusion, while the theoretical calculations suggest a much smaller length, practical considerations and the specific conditions of the gas evacuation process could lead to different interpretations. Always ensure to check the assumptions made in such problems!
