To determine the amount of heat supplied to the gas in the cylindrical container, we need to consider a few key factors: the work done by the gas as the piston moves, the properties of the gas, and the relationship between heat, work, and internal energy. Let's break this down step by step.
Understanding the System
We have a cylindrical container filled with oxygen, which behaves as an ideal gas under the given conditions. The piston has a mass of 50 kg and an area of cross-section of 100 cm² (which is 0.01 m² when converted to square meters). The atmospheric pressure is given as 100 kPa, and we know the gravitational acceleration is 10 m/s².
Calculating the Force on the Piston
The force exerted by the piston due to its weight can be calculated using the formula:
- Force (F) = mass (m) × gravitational acceleration (g)
Substituting the values:
- F = 50 kg × 10 m/s² = 500 N
Finding the Pressure Exerted by the Piston
The pressure exerted by the piston can be calculated using the formula:
- Pressure (P) = Force (F) / Area (A)
Using the area of the piston:
- P = 500 N / 0.01 m² = 50,000 Pa = 50 kPa
Total Pressure in the Cylinder
The total pressure inside the cylinder is the sum of the atmospheric pressure and the pressure exerted by the piston:
- Total Pressure (P_total) = Atmospheric Pressure + Pressure from Piston
Calculating this gives us:
- P_total = 100 kPa + 50 kPa = 150 kPa = 150,000 Pa
Work Done by the Gas
As the piston moves out through a distance of 20 cm (0.2 m), we can calculate the work done by the gas using the formula:
- Work (W) = Pressure (P) × Change in Volume (ΔV)
The change in volume can be calculated as:
- ΔV = Area × Distance = 0.01 m² × 0.2 m = 0.002 m³
Now substituting the values into the work formula:
- W = 150,000 Pa × 0.002 m³ = 300 J
Heat Supplied to the Gas
In a closed system where the gas is heated and does work, the first law of thermodynamics states:
Where ΔQ is the heat supplied, ΔU is the change in internal energy, and W is the work done. For an ideal gas, the change in internal energy can be related to the temperature change, but since we are not given specific temperature changes or heat capacities, we can assume that all the work done goes into heating the gas in this scenario.
Thus, the heat supplied to the gas can be approximated as:
In summary, the amount of heat supplied to the gas as the piston moves out through a distance of 20 cm is approximately 300 Joules. This calculation assumes that all the work done by the gas translates directly into heat energy, which is a reasonable approximation for this scenario.
