Problem 20 - College Physics

Question

\(\cdot\) In an effort to stay awake for an all-night study session, a student makes a cup of coffee by first placing a 200.0 \(\mathrm{W}\) electric immersion heater in 0.320 \(\mathrm{kg}\) of water. (a) How much heat must be added to the water to raise its temperature from \(20.0^{\circ} \mathrm{C}\) to \(80.0^{\circ} \mathrm{C} ?\) (b) How much time is required if all of the heater's power goes into heating the water?

Step-by-Step Solution

Answer

The heat required is 80,371.2 J, and it takes about 6.70 minutes to heat the water.

Step 1: Understanding the Problem

We'll divide the problem into two parts. First, calculate the heat needed to increase the water's temperature. Then, calculate the time required for the heater to provide this amount of energy.

Step 2: Calculate the Heat Energy Required

We need to calculate the heat energy using the formula: \[ q = mc\Delta T \] where:- \( m = 0.320 \, \text{kg} \)- \( c \) is the specific heat capacity of water, approximately \( 4.186 \, \text{J/g}^\circ\text{C} \) or \( 4186 \, \text{J/kg}^\circ\text{C} \)- \( \Delta T = 80.0^{\circ}\text{C} - 20.0^{\circ}\text{C} = 60.0^{\circ}\text{C} \)Substitute these values into the formula:\[ q = 0.320 \, \text{kg} \times 4186 \, \text{J/kg}^\circ\text{C} \times 60.0^{\circ}\text{C} = 80,371.2 \, \text{J} \]

Step 3: Calculate the Time Required

We use the formula for power: \[ P = \frac{q}{t} \] Rearranging for time \( t \):\[ t = \frac{q}{P} \]Substitute the values \( q = 80,371.2 \, \text{J} \) and \( P = 200.0 \, \text{W} \):\[ t = \frac{80,371.2 \, \text{J}}{200.0 \, \text{W}} = 401.856 \, \text{s} \]

Step 4: Converting Seconds to Minutes

Since the result is in seconds, we'll convert it to minutes for practicality:\[ t = \frac{401.856 \, \text{s}}{60 \, \text{s/min}} \approx 6.70 \, \text{minutes} \]

Key Concepts

Specific Heat Capacity

Specific heat capacity is a measure of how much thermal energy is required to raise the temperature of a given mass of a substance by one degree Celsius. Not all substances heat up at the same rate.
This is why specific heat capacity is crucial in understanding how different materials react to thermal energy. Imagine holding a metal spoon and a wooden spoon over a flame.
The metal spoon heats up faster than the wooden spoon.
This difference is due to their specific heat capacities. Water has a relatively high specific heat capacity, approximately 4186 J/kg°C, meaning it takes a lot of energy to change its temperature.

• Units: Joules per kilogram per degree Celsius (J/kg°C)
• Varies with material
• Important for designing heating and cooling systems, like the coffee heater example

Electric Power

Electric power is the rate at which electrical energy is transferred by an electric circuit.
It is measured in watts (W).
One watt corresponds to the consumption of one joule of energy per second. In the context of the heated cup of coffee, the immersion heater utilizes electric power to convert electrical energy into thermal energy. This conversion is key to increasing the water's temperature. Electric power is given by the formula:\[ P = \frac{E}{t} \]Where:

• \(P\) is power (in watts)
• \(E\) is energy (in joules)
• \(t\) is time (in seconds)

This means a 200 W heater can provide 200 Joules of energy every second, efficiently warming up water for our student’s coffee.

Thermal Energy

Thermal energy is the total internal kinetic energy of particles in a substance. When you heat water to make coffee, you're transferring thermal energy from the heater to the water molecules. The more thermal energy is transferred, the more the particles vibrate, increasing the water's temperature. In practical scenarios like boiling water, understanding thermal energy helps in predicting how long a process will take or how much heat energy is required.
Mathematically, this is represented as:\[ q = mc\Delta T \]Where:

• \(q\) is the thermal energy (in joules)
• \(m\) is the mass of the water (in kilograms)
• \(c\) is the specific heat capacity
• \(\Delta T\) is the change in temperature (in °C)

This calculation is vital when you need to know how much energy your immersion heater should provide.

Temperature Change

Temperature change signifies the difference between the initial and final temperatures of a substance when energy is added or removed. It is a critical factor in calculating the thermal energy needed and in assessing how quickly a process like heating water can happen. In our coffee example, the water's temperature is raised from 20°C to 80°C, giving a temperature change \(\Delta T\) of 60°C. Understanding temperature change helps in:

• Designing heating systems
• Predicting energy consumption
• Adjusting heat transfer according to needs

Whether you're a student attempting an all-nighter or designing industrial systems, grasping temperature change ensures effective thermal management. Combining this understanding with specific heat capacity forms the basis for effective problem-solving in physics.