To determine the mass of mercury needed to fill the flask at –35°C, we first need to understand how the volume of both the glass flask and the mercury changes with temperature. The key here is to apply the concepts of thermal expansion for both materials.
Understanding Volume Expansion
When materials are heated or cooled, they expand or contract. This is quantified by the coefficient of volume expansion, which tells us how much the volume of a substance changes per degree change in temperature. For our case:
- The coefficient of volume expansion for glass is 9.0 × 10-6 /°C.
- The coefficient of volume expansion for mercury is 1.8 × 10-4 /°C.
Calculating Volume Change of the Flask
First, we need to find out how much the volume of the glass flask changes when the temperature drops from 100°C to –35°C. The temperature change (ΔT) is:
ΔT = –35°C – 100°C = –135°C
Now, we can calculate the change in volume (ΔV) of the glass flask using the formula:
ΔV = V₀ × β × ΔT
Where:
- V₀ is the initial volume of the flask (which we can find from the mass of mercury at 100°C).
- β is the coefficient of volume expansion for glass.
- ΔT is the change in temperature.
Since we know the mass of mercury at 100°C is 901 g, we can find its volume using the density of mercury, which is approximately 13.6 g/cm³:
V₀ = mass/density = 901 g / 13.6 g/cm³ ≈ 66.2 cm³
Now, substituting the values into the volume change formula:
ΔV = 66.2 cm³ × (9.0 × 10-6 /°C) × (–135°C) ≈ –0.000803 cm³
This means the volume of the flask decreases by approximately 0.000803 cm³ when cooled to –35°C.
Calculating the New Volume of the Flask
The new volume of the flask (V_f) at –35°C is:
V_f = V₀ + ΔV = 66.2 cm³ – 0.000803 cm³ ≈ 66.1992 cm³
Calculating the Mass of Mercury at –35°C
Next, we need to find out how much mercury is required to fill this new volume at –35°C. Since the volume of mercury also changes with temperature, we need to calculate the volume of mercury at 100°C that corresponds to the volume at –35°C.
Using the volume expansion formula for mercury:
ΔV_mercury = V₀_mercury × β_mercury × ΔT_mercury
We can rearrange this to find the volume of mercury at –35°C:
V_f_mercury = V₀_mercury / (1 + β_mercury × ΔT_mercury)
Where:
- V₀_mercury is the volume of mercury at 100°C (66.2 cm³).
- β_mercury is the coefficient of volume expansion for mercury.
- ΔT_mercury is the change in temperature for mercury (–135°C).
Substituting the values:
V_f_mercury = 66.2 cm³ / (1 + (1.8 × 10-4 /°C) × (–135°C))
Calculating the denominator:
1 + (1.8 × 10-4 × –135) ≈ 1 – 0.0243 ≈ 0.9757
Now, substituting back:
V_f_mercury ≈ 66.2 cm³ / 0.9757 ≈ 67.8 cm³
Finding the Mass of Mercury
Finally, we can find the mass of mercury needed at –35°C using the density of mercury:
mass_mercury = V_f_mercury × density_mercury
mass_mercury ≈ 67.8 cm³ × 13.6 g/cm³ ≈ 921.2 g
Thus, the mass of mercury needed to fill the flask at –35°C is approximately 921.2 g.
