An electric heater of 800 watts is placed in a vessel containing 5 liters-of water at 20^circ C (C_(w) =4.19kJ/(kgcdot grad) Determine how long it takes for the water to heat to a boiling point of 100^circ C Neglect the heat loss of the vessel to the environment.

To determine how long it takes for the water to heat to a boiling point of $100^{\circ }C$, we can use the formula:<br /><br />\[ Q = mc\Delta T \]<br /><br />Where:<br />- \( Q \) is the heat energy required to heat the water<br />- \( m \) is the mass of the water<br />- \( c \) is the specific heat capacity of water<br />- \( \Delta T \) is the change in temperature<br /><br />Given:<br />- Power of the electric heater = 800 watts<br />- Volume of water = 5 liters<br />- Initial temperature of water = $20^{\circ }C$<br />- Final temperature of water = $100^{\circ }C$<br />- Specific heat capacity of water (\( C_w \)) = 4.19 kJ/(kg·°C)<br /><br />First, we need to convert the volume of water to mass. Since the density of water is approximately 1 kg/L, the mass of the water is 5 kg.<br /><br />Now, we can calculate the heat energy required to heat the water:<br /><br />\[ Q = 5 \, \text{kg} \times 4.19 \, \text{kJ/(kg·°C)} \times (100^{\circ }C - 20^{\circ }C) \]<br /><br />\[ Q = 5 \, \text{kg} \times 4.19 \, \text{kJ/(kg·°C)} \times 80^{\circ }C \]<br /><br />\[ Q = 167.6 \, \text{kJ} \]<br /><br />Next, we can calculate the time it takes for the water to heat to the boiling point using the formula:<br /><br />\[ t = \frac{Q}{P} \]<br /><br />Where:<br />- \( t \) is the time in seconds<br />- \( Q \) is the heat energy required in joules<br />- \( P \) is the power of the electric heater in watts<br /><br />Since 1 kJ = 1000 J, we need to convert the heat energy from kJ to J:<br /><br />\[ Q = 167.6 \, \text{kJ} \times 1000 \, \text{J/kJ} = 167600 \, \text{J} \]<br /><br />Now, we can calculate the time:<br /><br />\[ t = \frac{167600 \, \text{J}}{800 \, \text{W}} \]<br /><br />\[ t = 209.5 \, \text{seconds} \]<br /><br />Therefore, it takes approximately 209.5 seconds for the water to heat to a boiling point of $100^{\circ }C$.