Question
A mercury thermometer is constructed as shown. The capillary tube has a diameter of 0.0023 cm, and the bulb has a diameter of 0.27 cm. Neglecting the expansion of the glass, find the change in height of the mercury column for a temperature change of 20 ◦C. The volume expansion coefficient for mercury is 0.000182 (◦C)−1. Answer in units of cm.
Answers
GPT-4o mini
To find the change in height of the mercury column in the thermometer as a result of a temperature change, we need to consider the volume expansion of the mercury due to the temperature change and how that relates to the height change in the capillary tube.
### Step 1: Calculate the initial volume of mercury in the bulb
The volume \( V \) of the mercury in the bulb can be approximated using the formula for the volume of a sphere. However, given the size, we can treat the bulb as a cylinder or use the formula for volume of a sphere as follows:
For a bulb with diameter \( d_b = 0.27 \) cm, its radius \( r_b \) is:
\[
r_b = \frac{d_b}{2} = \frac{0.27 \, \text{cm}}{2} = 0.135 \, \text{cm}
\]
The volume \( V_b \) of the bulb is:
\[
V_b = \frac{4}{3} \pi r_b^3 \approx \frac{4}{3} \pi (0.135 \, \text{cm})^3
\]
Calculating this gives:
\[
V_b \approx \frac{4}{3} \pi (0.135)^3 \approx \frac{4}{3} \pi (0.002460375) \approx 0.010332 \, \text{cm}^3
\]
### Step 2: Calculate the change in volume of mercury
The change in volume \( \Delta V \) due to temperature change \( \Delta T = 20 \, \text{°C} \) can be calculated using the formula for volume expansion:
\[
\Delta V = V \beta \Delta T
\]
where \( \beta = 0.000182 \, (\text{°C})^{-1} \).
Thus,
\[
\Delta V = V_b \times \beta \times \Delta T
\]
Substituting the values we calculated:
\[
\Delta V = 0.010332 \, \text{cm}^3 \times 0.000182 \, (\text{°C})^{-1} \times 20 \, \text{°C}
\]
Calculating this step by step:
\[
\Delta V \approx 0.010332 \times 0.000182 \times 20 \approx 0.00003761952 \, \text{cm}^3
\]
### Step 3: Calculate the change in height of the mercury column
The change in height \( \Delta h \) in the capillary tube can be determined from the relationship between volume and cross-sectional area:
\[
\Delta V = A \Delta h
\]
Where \( A \) is the cross-sectional area of the capillary tube. The area \( A \) for the capillary tube with diameter \( d_c = 0.0023 \) cm is:
\[
A = \pi \left(\frac{d_c}{2}\right)^2 = \pi \left(\frac{0.0023}{2}\right)^2
\]
Calculating the area:
\[
A = \pi \left(0.00115\right)^2 \approx \pi \times 0.0000013225 \approx 0.000004162 \, \text{cm}^2
\]
### Step 4: Determine the change in height
Now, we can solve for \( \Delta h \):
\[
\Delta h = \frac{\Delta V}{A}
\]
Substituting known values:
\[
\Delta h = \frac{0.00003761952 \, \text{cm}^3}{0.000004162 \, \text{cm}^2} \approx 9.034 \, \text{cm}
\]
Thus, the change in height of the mercury column for a temperature change of 20 °C is approximately:
\[
\Delta h \approx 9.03 \, \text{cm}
\]
### Final Answer
The change in height of the mercury column is \( \Delta h \approx 9.03 \, \text{cm} \).
### Step 1: Calculate the initial volume of mercury in the bulb
The volume \( V \) of the mercury in the bulb can be approximated using the formula for the volume of a sphere. However, given the size, we can treat the bulb as a cylinder or use the formula for volume of a sphere as follows:
For a bulb with diameter \( d_b = 0.27 \) cm, its radius \( r_b \) is:
\[
r_b = \frac{d_b}{2} = \frac{0.27 \, \text{cm}}{2} = 0.135 \, \text{cm}
\]
The volume \( V_b \) of the bulb is:
\[
V_b = \frac{4}{3} \pi r_b^3 \approx \frac{4}{3} \pi (0.135 \, \text{cm})^3
\]
Calculating this gives:
\[
V_b \approx \frac{4}{3} \pi (0.135)^3 \approx \frac{4}{3} \pi (0.002460375) \approx 0.010332 \, \text{cm}^3
\]
### Step 2: Calculate the change in volume of mercury
The change in volume \( \Delta V \) due to temperature change \( \Delta T = 20 \, \text{°C} \) can be calculated using the formula for volume expansion:
\[
\Delta V = V \beta \Delta T
\]
where \( \beta = 0.000182 \, (\text{°C})^{-1} \).
Thus,
\[
\Delta V = V_b \times \beta \times \Delta T
\]
Substituting the values we calculated:
\[
\Delta V = 0.010332 \, \text{cm}^3 \times 0.000182 \, (\text{°C})^{-1} \times 20 \, \text{°C}
\]
Calculating this step by step:
\[
\Delta V \approx 0.010332 \times 0.000182 \times 20 \approx 0.00003761952 \, \text{cm}^3
\]
### Step 3: Calculate the change in height of the mercury column
The change in height \( \Delta h \) in the capillary tube can be determined from the relationship between volume and cross-sectional area:
\[
\Delta V = A \Delta h
\]
Where \( A \) is the cross-sectional area of the capillary tube. The area \( A \) for the capillary tube with diameter \( d_c = 0.0023 \) cm is:
\[
A = \pi \left(\frac{d_c}{2}\right)^2 = \pi \left(\frac{0.0023}{2}\right)^2
\]
Calculating the area:
\[
A = \pi \left(0.00115\right)^2 \approx \pi \times 0.0000013225 \approx 0.000004162 \, \text{cm}^2
\]
### Step 4: Determine the change in height
Now, we can solve for \( \Delta h \):
\[
\Delta h = \frac{\Delta V}{A}
\]
Substituting known values:
\[
\Delta h = \frac{0.00003761952 \, \text{cm}^3}{0.000004162 \, \text{cm}^2} \approx 9.034 \, \text{cm}
\]
Thus, the change in height of the mercury column for a temperature change of 20 °C is approximately:
\[
\Delta h \approx 9.03 \, \text{cm}
\]
### Final Answer
The change in height of the mercury column is \( \Delta h \approx 9.03 \, \text{cm} \).