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:** $502.32\text{ kJ}$

To determine the heat energy required to raise the temperature of water from \(20^\circ C\) to \(80^\circ C\), we'll use the formula:

\[
Q = m \times c \times \Delta T
\]

**Given:**
- Mass (\(m\)) = 2 kg
- Specific heat capacity of water (\(c\)) = 4186 J/(kg·K)
- Initial temperature (\(T_{\text{initial}}\)) = \(20^\circ C\)
- Final temperature (\(T_{\text{final}}\)) = \(80^\circ C\)

**Step 1: Calculate the change in temperature (\(\Delta T\))**

\[
\Delta T = T_{\text{final}} - T_{\text{initial}} = 80^\circ C - 20^\circ C = 60 \, ^\circ C
\]

Since the temperature difference is the same in Kelvin and Celsius:

\[
\Delta T = 60 \, \text{K}
\]

**Step 2: Calculate the heat energy (\(Q\))**

Plugging the values into the formula:

\[
Q = 2 \, \text{kg} \times 4186 \, \frac{\text{J}}{\text{kg} \cdot \text{K}} \times 60 \, \text{K}
\]

\[
Q = 2 \times 4186 \times 60
\]

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

**Step 3: Convert Joules to Kilojoules**

\[
Q = \frac{502,\!320}{1000} = 502.32 \, \text{kJ}
\]

**Final Answer:**

The heat energy required is:

\[
\boxed{502320\ \text{J or } 502.32\ \text{kJ}}
\]

<think>
Okay, so I need to figure out how much heat energy is required to raise the temperature of 2 kg of water from 20°C to 80°C. The problem gives me the specific heat capacity of water, which is 4186 J/(kg·K), and the formula Q = m*c*ΔT. Alright, let me start by recalling what each of these variables stands for. 

First, m is the mass of the substance, which in this case is water. The mass given here is 2 kg. Then c is the specific heat capacity, which tells us how much energy is needed to raise the temperature of 1 kg of the substance by 1 degree Kelvin. Since the temperature is given in Celsius, I remember that the change in temperature (ΔT) in Celsius is the same as in Kelvin, so that's straightforward. The ΔT here would be the final temperature minus the initial temperature. 

Let me write down the values first:

- Mass (m) = 2 kg
- Specific heat capacity (c) = 4186 J/(kg·K)
- Initial temperature (T_initial) = 20°C
- Final temperature (T_final) = 80°C

So, the change in temperature ΔT is T_final - T_initial. Let me calculate that first. 

ΔT = 80°C - 20°C = 60°C. Since it's a temperature difference, this is equivalent to 60 K as well because the size of a degree Celsius is the same as a Kelvin. 

Now, plugging all the values into the formula Q = m*c*ΔT:

Q = 2 kg * 4186 J/(kg·K) * 60 K

Hmm, let me compute the numerical part step by step. 

First, multiply the mass and the specific heat capacity:

2 kg * 4186 J/(kg·K) = 8372 J/K

Then multiply that result by the temperature change:

8372 J/K * 60 K = ?

Let me do that multiplication. 8372 * 60. Hmm, 8000*60 is 480,000, and 372*60 is 22,320. Adding those together: 480,000 + 22,320 = 502,320 J. Wait, but let me check that again because 8372 * 60:

Alternatively, breaking it down:

8000 * 60 = 480,000

300 * 60 = 18,000

72 * 60 = 4,320

Adding those three parts: 480,000 + 18,000 = 498,000; 498,000 + 4,320 = 502,320 J. 

Wait, but 8372 * 60:

Alternatively, 8372 * 6 * 10. Let's see:

8372 * 6: 

8000*6 = 48,000

372*6 = 2,232

Total is 48,000 + 2,232 = 50,232

Then multiply by 10 gives 502,320 J. Yeah that's the same. 

So Q = 502,320 J. 

But the question asks for the answer in both joules and kilojoules. Since 1 kJ is 1000 J, to convert to kilojoules, divide by 1000:

502,320 J ÷ 1000 = 502.32 kJ. 

Let me just verify the formula again to make sure I didn't mix up anything. The formula is Q = m*c*ΔT, and I substituted the values correctly. 

Mass is in kg, which matches the units of specific heat (kg in the denominator). The temperature difference is in K, which matches as well. So the units should work out to J. 

Wait, let me check the units as well to confirm:

kg * J/(kg·K) * K = J. Yep, the kg cancels, the K cancels, leaving Joules. Perfect. 

So 502,320 J is correct, and 502.32 kJ. 

Hmm, but let me check my arithmetic again, just to be thorough. 

2 * 4186 is 8372. Then 8372 * 60:

Let me do 8372 * 60:

8372 * 60:

Let me compute 8372 * 6 first:

8,000 * 6 = 48,000

372 * 6 = 2,232

Adding them gives 48,000 + 2,232 = 50,232

Then multiply by 10 (because 60 is 6*10), so 50,232 * 10 = 502,320. That's correct.

Alternatively, using calculator-like steps:

8372 * 60:

Break 60 into 50 + 10:

8372*50 = 8372*5*10 = (41,860)*10 = 418,600

8372*10 = 83,720

Add them: 418,600 + 83,720 = 502,320. Same result. 

So, the calculation seems consistent. 

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

Wait, but perhaps the question expects rounding? Let me check the given numbers. The mass is exactly 2 kg, the temperatures are whole numbers (20 to 80), and the specific heat is given as 4186, which is precise. So maybe they want the exact figure. 

Alternatively, sometimes specific heat capacity of water is approximated as 4186 J/(kg·K), so perhaps we can present it as 502,320 J or 502.32 kJ. 

Alternatively, maybe the problem expects it written as 502320 J and 502.32 kJ. But let me check once again. 

Wait, 2 kg times 4186 is 8372, times 60 is indeed 502,320. So that's correct. 

Alternatively, maybe they want it expressed in a different way? Let me see:

Alternatively, the calculation steps:

Q = m * c * (T_final - T_initial)

Q = 2 kg * 4186 J/(kg·K) * (80°C - 20°C)

= 2 * 4186 *

To calculate the amount of heat energy required to raise the temperature of $2\text{ kg}$ of water from $20^\circ\text{C}$ to $80^\circ\text{C}$, we'll use the formula:
$$Q = m \cdot c \cdot ΔT, $$
where 
- $m$ is the mass of the substance ($2\text{ kg}$),
- $c$ is the specific heat capacity of water ($4186\frac{\text{J}}{\text{kg}\cdot\text{K}})$, and
- $\Delta T = 80^\circ\text{C} - 20 ^\circ\text{C}$ is the temperature change. 

Plugging in these values:
$$Q = (2 \text{ kg})(4186 \frac{\text{J}}{\text{kg}\cdot\text{K}})((80^\circ\text{C} - 20 ^\circ\text{C}))= 1000,320 \text{ J}.$$

Thus the amount of heat energy required is $\boxed{1000,320 \text{ J}}$ or about $\boxed{1.00 \text{ kJ}}$.