The specific heat of a substance varies as (4θ² + 2θ) x 10^-3 cal/g/^oC. Find out the amount of heat required to raise the temperature of 1 kg of substance from 10^oC to 20^oC.

To find the amount of heat required to raise the temperature of a substance from 10 °C to 20 °C, we need to use the concept of specific heat, which varies with temperature in this case. The specific heat of the substance is given by the formula (4θ² + 2θ) x 10⁻³ cal/g/°C, where θ represents the temperature in degrees Celsius. Since we are dealing with a mass of 1 kg, we will convert that to grams for our calculations, as specific heat is typically expressed in cal/g/°C.

Step-by-Step Calculation

1. **Convert mass to grams**: Since 1 kg is equal to 1000 grams, we will use 1000 g in our calculations.

2. **Determine the specific heat at different temperatures**: We need to calculate the specific heat at the initial temperature (10 °C) and the final temperature (20 °C).

• At θ = 10 °C:

  Specific heat, c(10) = (4(10)² + 2(10)) x 10⁻³ = (4(100) + 20) x 10⁻³ = (400 + 20) x 10⁻³ = 420 x 10⁻³ = 0.42 cal/g/°C

• At θ = 20 °C:

  Specific heat, c(20) = (4(20)² + 2(20)) x 10⁻³ = (4(400) + 40) x 10⁻³ = (1600 + 40) x 10⁻³ = 1640 x 10⁻³ = 1.64 cal/g/°C

3. **Calculate the heat required**: Since the specific heat changes with temperature, we can use the average specific heat over the temperature range from 10 °C to 20 °C to simplify our calculation. The average specific heat (c_avg) can be calculated as follows:

c_avg = (c(10) + c(20)) / 2 = (0.42 + 1.64) / 2 = 2.06 / 2 = 1.03 cal/g/°C

4. **Use the formula for heat**: The amount of heat (Q) required to raise the temperature can be calculated using the formula:

Q = m × c_avg × ΔT

Where:

• m = mass of the substance (1000 g)
• c_avg = average specific heat (1.03 cal/g/°C)
• ΔT = change in temperature (20 °C - 10 °C = 10 °C)

5. **Plug in the values**:

Q = 1000 g × 1.03 cal/g/°C × 10 °C = 10300 cal

Final Result

The total amount of heat required to raise the temperature of 1 kg of the substance from 10 °C to 20 °C is 10,300 calories.

This method illustrates how to handle varying specific heat and calculate the heat transfer required for temperature changes in a substance. If you have any further questions or need clarification on any part of the process, feel free to ask!