A 0.75–L bottle is cleaned, dried, and closed in a room where the air is 22oC and 44% relative humidity (that is, the water vapour in the air is 0.44 of the equilibrium vapour pressure at 22oC). The bottle is brought outside and stored at 0.0oC.
a) What mass of liquid water condenses inside the bottle at this temperature?
b) Would liquid water condense at 10oC?
(See Table 5.3, p.207, Silberberg for reference)
To answer these questions, we need to use the concept of saturation vapor pressure and the Clausius-Clapeyron equation. The saturation vapor pressure is the maximum amount of water vapor that air can hold at a given temperature. The Clausius-Clapeyron equation relates the saturation vapor pressure at two different temperatures.
Let's start by calculating the saturation vapor pressure at 22°C using the given relative humidity (RH). The equilibrium vapor pressure (EVP) is the saturation vapor pressure at a particular temperature multiplied by the RH. So, the EVP at 22°C is:
EVP = 0.44 * saturation vapor pressure at 22°C
To find the saturation vapor pressure at 22°C, we can use Table 5.3 in the Silberberg textbook.
Now, to find the mass of liquid water that condenses inside the bottle at 0.0°C (Question a), we need to compare the saturation vapor pressure at 0.0°C with the EVP at 22°C.
Next, we can calculate the saturation vapor pressure at 0.0°C using Table 5.3 in the Silberberg textbook.
Finally, we compare the saturation vapor pressure at 0.0°C with the EVP at 22°C to determine if condensation occurs. If the saturation vapor pressure at 0.0°C is greater than the EVP at 22°C, condensation will occur, and we can calculate the mass of water using the ideal gas law:
PV = nRT
Where:
P is the vapor pressure,
V is the volume of the container (0.75 L),
n is the number of moles of water vapor,
R is the ideal gas constant,
T is the temperature (0.0°C),
and rearranging the equation, we can solve for n (number of moles of water vapor).
To convert the number of moles to mass, you can use the molar mass of water and multiply it with the number of moles.
To answer Question b, we repeat the same procedure but this time, we compare the saturation vapor pressure at 10.0°C with the EVP at 22°C. If the saturation vapor pressure is lower than the EVP, condensation will not occur.
Please refer to Table 5.3 in the Silberberg textbook to obtain the necessary data for the calculations.