Introduction:
In this problem, we are given an open vessel containing 200mg of air at a temperature of 17°C. We need to determine the weight of air that would be expelled if the vessel is heated to a temperature of 117°C. To solve this problem, we will use Charles's Law, which states that the volume of a gas is directly proportional to its temperature in Kelvin.

Given:
Initial temperature (T1) = 17°C
Final temperature (T2) = 117°C
Initial weight of air (W1) = 200mg

Step 1: Convert temperatures to Kelvin:
The temperature must be in Kelvin for Charles's Law. We can convert Celsius to Kelvin by adding 273.15.
T1 = 17 + 273.15 = 290.15 K
T2 = 117 + 273.15 = 390.15 K

Step 2: Apply Charles's Law equation:
According to Charles's Law, V1/T1 = V2/T2, where V1 and V2 are the initial and final volumes respectively.
Since the vessel is open, the volume remains constant. Therefore, we can rewrite the equation as:
T1/T2 = V1/V2

Step 3: Calculate the final volume:
Since the volume is constant, V1/V2 = 1, and we can rewrite the equation as:
T1/T2 = 1

Step 4: Calculate the weight of expelled air:
Now, we can use the equation V2 = V1(T2/T1) to find the final volume.
Since the volume remains constant, the weight of air expelled will be directly proportional to the change in volume. Therefore, the weight of expelled air (W2) can be calculated as:
W2 = W1 * (V2/V1)

Step 5: Substitute the values:
W2 = 200mg * (1/1) = 200mg

Explanation:
When the vessel is heated to 117°C, the weight of air expelled will be 200mg. This is because the volume of the gas is directly proportional to its temperature when the pressure and the amount of gas remain constant. As the temperature increases, the gas molecules gain kinetic energy and move faster, causing the gas to expand. Since the volume of the vessel is constant, the gas has to escape, resulting in the expulsion of some air. Therefore, the weight of expelled air remains the same as the initial weight of air, which is 200mg.

Conclusion:
When the vessel is heated to 117°C, the weight of air expelled will be 200mg. This can be explained using Charles's Law, which states that the volume of a gas is directly proportional to its temperature in Kelvin.