Question

The specific heat of iron metal is \(0.450 \mathrm{~J} / \mathrm{g}-\mathrm{K}\). How many J of heat are necessary to raise the temperature of a 1.05-kg block of iron from \(25.0{ }^{\circ} \mathrm{C}\) to \(88.5^{\circ} \mathrm{C}\) ?

Short answer

The heat necessary to raise the temperature of a 1.05-kg block of iron from \(25.0{ }^{\circ} \mathrm{C}\) to \(88.5{ }^{\circ} \mathrm{C}\) can be calculated using the formula \(q = mc\Delta T\). Converting the mass to grams (1050 g) and calculating the change in temperature (63.5 K), we find that the heat absorbed is \(q = (1050 \mathrm{g})(0.450 \mathrm{J/g-K})(63.5 \mathrm{K}) = 29925 \mathrm{J}\). So, 29,925 J of heat are necessary.

Step-by-step solution

Convert the mass from kg to g

In order to convert the mass of the iron block from kg to g, we need to multiply by 1000 since there are 1000 g in 1 kg: mass (g) = 1.05 kg * 1000 (g/kg) = 1050 g

Calculate the change in temperature

ΔT is the difference between the final temperature (88.5°C) and the initial temperature (25.0°C). To calculate the change in temperature, subtract the initial temperature from the final temperature: ΔT = 88.5°C - 25.0°C = 63.5 K (Note: The change in temperature is expressed in Kelvin (K) since the specific heat capacity is given in J/g-K. However, the change in temperature is the same if expressed in Celsius degrees (°C). So, we can leave ΔT in K.)

Calculate the heat absorbed using the formula

Now that we have the mass in g and the change in temperature in K, we can use the formula to find the heat absorbed: q = mcΔT q = (1050 g)(0.450 J/g-K)(63.5 K) q = 29925 J So, 29,925 J of heat are necessary to raise the temperature of the 1.05-kg block of iron from 25.0°C to 88.5°C.

Key concepts

Heat Transfer Calculation

Heat transfer is the process of energy exchange due to a temperature difference. In calculations like this, understanding how to determine the amount of heat exchanged is crucial. The specific heat capacity is the key property that allows us to relate the mass, temperature change, and energy exchanged.

To perform a heat transfer calculation, use the equation:

• \( q = mc\Delta T \)
• \( q \) is the heat absorbed or released (in joules).
• \( m \) is the mass (in grams, once converted).
• \( c \) is the specific heat capacity (in J/g-K).
• \( \Delta T \) is the change in temperature (in K or °C).

In our exercise, we applied this formula to find out how much energy is needed to raise the temperature of iron. By plugging in the mass, specific heat, and temperature change, we calculated the heat required. Thus, solving \( q = (1050 \text{ g})(0.450 \text{ J/g-K})(63.5 \text{ K}) = 29925 \text{ J} \).

Understanding heat transfer calculations is helpful in many scientific applications and everyday situations, such as predicting energy use in heating systems.

Temperature Change

Temperature change, symbolized as \( \Delta T \), indicates the difference in the initial and final states of an object's temperature. In physics and chemistry, it represents the driving force behind heat transfer. Calculating temperature change is simple yet essential.

Here's how you can figure it out:

• Determine the final temperature (\( T_f \)).
• Identify the initial temperature (\( T_i \)).
• Subtract the initial from the final temperature: \( \Delta T = T_f - T_i \).

In the context of our problem, the initial temperature is 25.0°C, while the final is 88.5°C, leading to:
\( \Delta T = 88.5 - 25.0 = 63.5 K \) (Kelvin and Celsius changes are equivalent).

The calculated temperature change helps us understand how much energy iron requires to achieve this increase. Even though it seems straightforward, getting this step right is crucial when using formulas involving heat.

Mass Conversion

Converting mass between different units is a common task when performing calculations in physics and chemistry, especially when using formulae in science that expect specific units. In this exercise, converting the iron block's mass from kilograms to grams was needed, as specific heat capacity was provided in J/g-K.

To convert mass from kilograms to grams:

• Note that 1 kilogram equals 1000 grams.
• Multiply the number of kilograms by 1000 to convert to grams.

For our iron block:
Mass in kilograms = 1.05 kg
Convert to grams = 1.05 kg * 1000 g/kg = 1050 g.

Converting mass accurately is significant because incorrect units can lead to errors in calculations like heat transfer and others. Consistent unit usage ensures correct results and understanding of material properties.