R51 · Heat required to warm water (specific heat capacity)

To calculate the amount of heat energy (Q) required to raise the temperature of 2 kg of water from 20°C to 80°C, use the formula:

Q = m * c * ΔT

where:
m is the mass of the water in kilograms
c is the specific heat capacity of water (4186 J/(kg*K))
ΔT is the change in temperature (in Kelvin)

First, convert the temperatures from Celsius to Kelvin:
20°C = 20 + 273.15 = 293.15 K
80°C = 80 + 273.15 = 353.15 K

The change in temperature ΔT is:
ΔT = T2 - T1 = 353.15 K - 293.15 K = 60 K

Now, plug the values into the formula:

Q = m * c * ΔT
Q = 2 kg * 4186 J/(kg*K) * 60 K
Q = 501120 J

To convert joules to kilojoules, divide by 1000:
Q ≈ 501.12 kJ

To calculate the heat energy required, we'll use the formula:

Q = m * c * ΔT

where:
- Q is the heat energy,
- m is the mass of the water,
- c is the specific heat capacity of water,
- ΔT is the change in temperature.

Given:
- m = 2 kg,
- c = 4186 J/(kg*K),
- ΔT = 80°C - 20°C = 60 K (or °C, since the size of a degree Kelvin is the same as a degree Celsius).

Now, plug in the values:

Q = 2 kg * 4186 J/(kg*K) * 60 K
Q = 8372 * 60
Q = 502,320 J

To convert joules to kilojoules (kJ), divide by 1,000:

Q = 502,320 J / 1,000
Q = 502.32 kJ

So, the heat energy required is 502,320 joules or 502.32 kilojoules.

Here is the step-by-step calculation to determine the required heat energy:

### Given Values:
* **Mass ($m$):** $2 \text{ kg}$
* **Specific Heat Capacity of water ($c$):** $4186 \text{ J/(kg}\cdot\text{K)}$
* **Initial Temperature ($T_{initial}$):** $20^\circ\text{C}$
* **Final Temperature ($T_{final}$):** $80^\circ\text{C}$

### 1. Calculate the Change in Temperature ($\Delta T$):
$$\Delta T = T_{final} - T_{initial}$$
$$\Delta T = 80^\circ\text{C} - 20^\circ\text{C}$$
$$\Delta T = 60^\circ\text{C}$$

*(Note: Since the temperature difference is used, a change of $60^\circ\text{C}$ is equal to a change of $60 \text{ K}$.) *

### 2. Use the Heat Energy Formula:
$$Q = m \cdot c \cdot \Delta T$$

### 3. Substitute and Calculate (in Joules):
$$Q = (2 \text{ kg}) \cdot (4186 \text{ J/(kg}\cdot\text{K)}) \cdot (60 \text{ K})$$
$$Q = 2 \cdot 4186 \cdot 60 \text{ J}$$
$$Q = 4186 \cdot 120 \text{ J}$$
$$Q = 502,320 \text{ J}$$

### 4. Convert to Kilojoules (kJ):
Since $1 \text{ kJ} = 1000 \text{ J}$:
$$Q (\text{kJ}) = \frac{502,320 \text{ J}}{1000}$$
$$Q (\text{kJ}) = 502.32 \text{ kJ}$$

***

### Final Answer:
The heat energy required is:
* **In Joules:** $\mathbf{502,320 \text{ J}}$
* **In Kilojoules:** $\mathbf{502.32 \text{ kJ}}$

This problem requires using the heat energy equation $Q = m \cdot c \cdot \Delta T$.

### 1. Identify Given Values and Calculate $\Delta T$

*   **Mass ($m$):** $2 \text{ kg}$
*   **Specific Heat Capacity ($c$):** $4186 \text{ J}/(\text{kg}\cdot\text{K})$
*   **Initial Temperature ($T_i$):** $20^\circ\text{C}$
*   **Final Temperature ($T_f$):** $80^\circ\text{C}$

First, calculate the change in temperature ($\Delta T$). Since $\Delta T$ is a change in temperature, we can use the difference in Celsius or Kelvin:
$$\Delta T = T_f - T_i$$
$$\Delta T = 80^\circ\text{C} - 20^\circ\text{C}$$
$$\Delta T = 60^\circ\text{C} \quad (\text{or } 60 \text{ K})$$

### 2. Calculation of Heat Energy ($Q$)

Now, substitute the values into the formula:
$$Q = m \cdot c \cdot \Delta T$$
$$Q = (2 \text{ kg}) \cdot (4186 \text{ J}/(\text{kg}\cdot\text{K})) \cdot (60 \text{ K})$$

$$Q = 50,232 \text{ J}$$

### 3. Final Answer Conversion

**In Joules:**
The heat energy required is **$50,232 \text{ J}$**.

**In Kilojoules (kJ):**
To convert joules to kilojoules, divide by 1000 ($1 \text{ kJ} = 1000 \text{ J}$):
$$Q = \frac{50,232}{1000} \text{ kJ}$$
The heat energy required is **$50.232 \text{ kJ}$**.

**Solution:**

We need to calculate the heat energy required to raise the temperature of 2 kg of water from \(20^\circ\text{C}\) to \(80^\circ\text{C}\). 

Given:
- Mass (\(m\)) = 2 kg
- Specific heat capacity (\(c\)) = 4186 J/(kg·K)
- Initial temperature (\(T_1\)) = \(20^\circ\text{C}\)
- Final temperature (\(T_2\)) = \(80^\circ\text{C}\)

First, calculate the change in temperature (\(\Delta T\)):
\[
\Delta T = T_2 - T_1 = 80^\circ\text{C} - 20^\circ\text{C} = 60^\circ\text{C}
\]

Since temperature differences in Celsius and Kelvin are the same, \(\Delta T\) remains \(60\text{K}\).

Now, use the formula for heat energy:
\[
Q = m \times c \times \Delta T
\]
\[
Q = 2\,\text{kg} \times 4186\,\text{J/(kg·K)} \times 60\,\text{K}
\]

Calculate the values step by step:
\[
2 \times 4186 = 8372
\]
\[
8372 \times 60 = 502,320
\]

So,
\[
Q = 502,320\,\text{J}
\]

To convert Joules to Kilojoules:
\[
502,320\,\text{J} = \frac{502,320}{1000}\,\text{kJ} = 502.32\,\text{kJ}
\]

**Final Answer:**
\[
\boxed{Q = 502,\!320\,\text{J}} \quad \text{or} \quad \boxed{Q = 502.32\,\text{kJ}}
\]

This problem requires using the formula for heat energy transfer:
$$Q = m \cdot c \cdot \Delta T$$

Where:
*   $Q$ is the heat energy required (in Joules, J)
*   $m$ is the mass (in kg)
*   $c$ is the specific heat capacity (in $\text{J}/(\text{kg}\cdot\text{K})$)
*   $\Delta T$ is the change in temperature ($\text{T}_{\text{final}} - \text{T}_{\text{initial}}$) (in K or $^\circ\text{C}$)

---

### 1. Identify Given Values and Calculate $\Delta T$

*   Mass ($m$) = $2 \text{ kg}$
*   Specific Heat Capacity ($c$) = $4186 \text{ J}/(\text{kg}\cdot\text{K})$
*   Initial Temperature ($\text{T}_{\text{initial}}$) = $20^\circ\text{C}$
*   Final Temperature ($\text{T}_{\text{final}}$) = $80^\circ\text{C}$

**Calculate the change in temperature ($\Delta T$):**
$$\Delta T = 80^\circ\text{C} - 20^\circ\text{C}$$
$$\Delta T = 60^\circ\text{C}$$

*(Note: Since we are calculating a temperature difference, $60^\circ\text{C}$ is equivalent to $60 \text{ K}$.)*

### 2. Calculation (Using the Formula)

Substitute the values into the equation:
$$Q = m \cdot c \cdot \Delta T$$
$$Q = (2 \text{ kg}) \cdot (4186 \text{ J}/(\text{kg}\cdot\text{K})) \cdot (60 \text{ K})$$

$$Q = 502,320 \text{ J}$$

### 3. Final Answer Conversion

**In Joules ($\text{J}$):**
$$Q = 502,320 \text{ J}$$

**In Kilojoules ($\text{kJ}$):**
(Since $1 \text{ kJ} = 1000 \text{ J}$, divide the result by 1000)
$$Q = \frac{502,320}{1000} \text{ kJ}$$
$$Q = 502.32 \text{ kJ}$$

---
The heat energy required is **$502,320 \text{ J}$** or **$502.32 \text{ kJ}$**.

To find the heat energy required, we will use the formula provided and follow these steps:

### 1. Identify the given values:
*   **Mass ($m$):** $2\text{ kg}$
*   **Specific heat capacity ($c$):** $4186\text{ J/(kg}\cdot\text{K)}$
*   **Initial temperature ($T_i$):** $20^\circ\text{C}$
*   **Final temperature ($T_f$):** $80^\circ\text{C}$

### 2. Calculate the change in temperature ($\Delta T$):
$$\Delta T = T_f - T_i$$
$$\Delta T = 80^\circ\text{C} - 20^\circ\text{C} = 60^\circ\text{C}$$
*(Note: A change of $60^\circ\text{C}$ is equivalent to a change of $60\text{ K}$)*

### 3. Calculate the heat energy ($Q$):
Using the formula $Q = m \cdot c \cdot \Delta T$:

$$Q = 2\text{ kg} \times 4186\text{ J/(kg}\cdot\text{K)} \times 60\text{ K}$$
$$Q = 8372 \times 60$$
$$Q = 502,320\text{ J}$$

### 4. Convert to kilojoules (kJ):
To convert Joules to kilojoules, divide by $1,000$:
$$502,320\text{ J} \div 1,000 = 502.32\text{ kJ}$$

***

**Final Answer:**
*   **In Joules:** $502,320\text{ J}$
*   **In Kilojoules:** $502.32\text{ kJ}$