A glass flask of volume 100 extcm3 is filled to the brim with liquid whose cubical coefficient of expansion is 1 imes10⁻³ ext per ^. The flask and its liquid content are originally at 20^. Determine

Here are the solutions to questions 10, 11, 12, and 13.

10. A glass flask of volume $100\text{cm}^3$ is filled to the brim with liquid whose cubical coefficient of expansion is $1\times10^{-3} \text{ per } ^\circ\text{C}$. The flask and its liquid content are originally at $20^\circ\text{C}$. Determine the volume of liquid which will overflow upon heating the flask to $50^\circ\text{C}$ if the coefficient of linear expansion of glass is $8\times10^{-6} \text{ per } ^\circ\text{C}$.

Step 1: Identify the given values and calculate the change in temperature.
Initial volume, $V_0 = 100 \text{ cm}^3$.
Cubical coefficient of expansion of liquid, $\gamma_{\text{liquid}} = 1 \times 10^{-3} \text{ per } ^\circ\text{C}$.
Linear coefficient of expansion of glass, $\alpha_{\text{glass}} = 8 \times 10^{-6} \text{ per } ^\circ\text{C}$.
Initial temperature, $T_1 = 20^\circ\text{C}$.
Final temperature, $T_2 = 50^\circ\text{C}$.
Change in temperature, $\Delta T = T_2 - T_1 = 50^\circ\text{C} - 20^\circ\text{C} = 30^\circ\text{C}$.

Step 2: Calculate the cubical coefficient of expansion for glass.
The cubical coefficient of expansion for a solid is approximately three times its linear coefficient of expansion:

$$\gamma_{\text{glass}} = 3 \alpha_{\text{glass}} = 3 \times (8 \times 10^{-6} \text{ per } ^\circ\text{C}) = 24 \times 10^{-6} \text{ per } ^\circ\text{C}$$

Step 3: Calculate the change in volume for the liquid and the flask.
The change in volume is given by $\Delta V = V_0 \gamma \Delta T$.
For the liquid:

$$\Delta V_{\text{liquid}} = V_0 \gamma_{\text{liquid}} \Delta T = (100 \text{ cm}^3) \times (1 \times 10^{-3} \text{ per } ^\circ\text{C}) \times (30^\circ\text{C})$$ $$\Delta V_{\text{liquid}} = 3 \text{ cm}^3$$

For the flask:

$$\Delta V_{\text{flask}} = V_0 \gamma_{\text{glass}} \Delta T = (100 \text{ cm}^3) \times (24 \times 10^{-6} \text{ per } ^\circ\text{C}) \times (30^\circ\text{C})$$ $$\Delta V_{\text{flask}} = 0.072 \text{ cm}^3$$

Step 4: Determine the volume of liquid that overflows.
The volume of liquid that overflows is the difference between the expansion of the liquid and the expansion of the flask.

$$\Delta V_{\text{overflow}} = \Delta V_{\text{liquid}} - \Delta V_{\text{flask}}$$ $$\Delta V_{\text{overflow}} = 3 \text{ cm}^3 - 0.072 \text{ cm}^3$$ $$\Delta V_{\text{overflow}} = 2.928 \text{ cm}^3$$

The volume of liquid that will overflow is $\boxed{2.928 \text{ cm}^3}$.

11. $500\text{cm}^3$ of water is to be heated from room temperature ($28^\circ\text{C}$) to $100^\circ\text{C}$ in order to prepare hot cup of coffee. (a) What is the minimum heat required? (b) How long will it take to heat the water with a $1000\text{W}$ heating coil which has a heating efficiency of $70\%$?

Given constants:

• Density of water, $\rho_{\text{water}} = 1 \text{ g/cm}^3 = 1000 \text{ kg/m}^3$.
• Specific heat capacity of water, $c_{\text{water}} = 4186 \text{ J/kg}^\circ\text{C}$.