A vertical glass tube of length L = 1.280 000 m is half filled with a liquid at `20.000 000 ^@C` .How much will the height of the liquid column change when the tube and liquid are healted to 30.000 000 `""^@C` ? Use coefficients `alpha_("glass") = 2.000 000 xx 10^(-5)//K and beta _("liquid")=4.000 000 xx10^(-5)//K`.

Answer

To solve the problem, we need to determine the change in height of the liquid column in the glass tube when both the tube and the liquid are heated from 20°C to 30°C.

### Step-by-Step Solution:

1. **Identify Initial Conditions:**
- Length of the glass tube, \( L = 1.280 \, \text{m} \)
- Initial temperature, \( T_i = 20.000 \, ^\circ C \)
- Final temperature, \( T_f = 30.000 \, ^\circ C \)
- Coefficient of linear expansion of glass, \( \alpha_{\text{glass}} = 2.000 \times 10^{-5} \, \text{K}^{-1} \)
- Coefficient of volume expansion of liquid, \( \beta_{\text{liquid}} = 4.000 \times 10^{-5} \, \text{K}^{-1} \)

2. **Calculate the Change in Temperature:**
\[ \Delta T = T_f - T_i = 30.000 - 20.000 = 10.000 \, ^\circ C \]

3. **Calculate the Initial Height of the Liquid Column:**
Since the tube is half-filled with liquid:
\[ L_i = \frac{L}{2} = \frac{1.280}{2} = 0.640 \, \text{m} \]

4. **Calculate the Change in Length of the Liquid Column:**
The change in volume of the liquid due to temperature change can be calculated using the formula:
\[ \Delta L_{\text{liquid}} = L_i \cdot \beta_{\text{liquid}} \cdot \Delta T \]
Substituting the values:
\[ \Delta L_{\text{liquid}} = 0.640 \cdot (4.000 \times 10^{-5}) \cdot 10 = 0.640 \cdot 4.000 \times 10^{-4} = 0.000256 \, \text{m} \]

5. **Calculate the Change in Length of the Glass Tube:**
The change in length of the glass tube can be calculated using the formula:
\[ \Delta L_{\text{glass}} = L \cdot \alpha_{\text{glass}} \cdot \Delta T \]
Substituting the values:
\[ \Delta L_{\text{glass}} = 1.280 \cdot (2.000 \times 10^{-5}) \cdot 10 = 1.280 \cdot 2.000 \times 10^{-4} = 0.000256 \, \text{m} \]

6. **Calculate the Final Height of the Liquid Column:**
The final height of the liquid column \( L_f \) can be calculated as:
\[ L_f = L_i + \Delta L_{\text{liquid}} - \Delta L_{\text{glass}} \]
Substituting the values:
\[ L_f = 0.640 + 0.000256 - 0.000256 = 0.640 \, \text{m} \]

7. **Calculate the Change in Height of the Liquid Column:**
The change in height of the liquid column is:
\[ \Delta h = L_f - L_i = 0.640 - 0.640 = 0.000 \, \text{m} \]

### Final Result:
The height of the liquid column does not change when the tube and liquid are heated from 20°C to 30°C. Therefore, the change in height of the liquid column is:
\[ \Delta h = 0.000 \, \text{m} \]