# Specific heat, latent heat and Carnot efficiency calculator

> Heat energy from Q = mcΔT (solve for heat, mass, final temperature or specific heat), latent heat Q = mL for melting or boiling, and Carnot efficiency.

Interactive version: https://www.calcopenly.com/science/heat-transfer-calculator
Subject: Science calculators

Warming or cooling a material takes heat Q = mcΔT, where m is the mass, c the specific heat capacity and ΔT the temperature change; the calculator solves for any one of heat, mass, final temperature or specific heat. Melting or boiling takes Q = mL at constant temperature, where L is the latent heat. The Carnot mode gives the upper limit on any heat engine's efficiency, 1 − Tc/Th with both temperatures in kelvin.

The default, 1 kg of water heated from 20 °C to 100 °C, needs 334,880 J (0.093 kWh), what a 2 kW kettle delivers in 2 minutes 47 seconds with no losses. Boiling that water away takes a further 2,256 kJ, almost seven times as much.

Specific heats are room-temperature values from OpenStax University Physics (Table 1.3). In reality c varies with temperature, and the calculation assumes no melting or boiling between the two temperatures.

## Inputs

- **Calculate** (options: Temperature change, Phase change, Carnot efficiency)
- **Solve for** (options: Heat, Mass, Final temperature, Specific heat)
- **Material** (options: Water (liquid, 15 °C), Ice (average −50 to 0 °C), Aluminium, Copper, Iron / steel, Lead, Silver, Gold, Glass, Concrete / granite, Wood, Ethanol, Mercury, Human body (average), Enter specific heat)
- **Specific heat capacity**
- **Heat added (negative if removed)**
- **Mass**
- **Initial temperature**
- **Final temperature**
- **Solve for** (options: Heat, Mass)
- **Substance** (options: Water, Ethanol, Nitrogen, Lead, Enter latent heat)
- **Change** (options: Melting / freezing, Boiling / condensing)
- **Latent heat**
- **Show heat in** (options: J, kJ, kcal, BTU)
- **Show temperatures in** (options: °C, °F, K)
- **Hot reservoir temperature**
- **Cold reservoir temperature**
- **Heat taken from the hot side**

## Results

- Heat (J) — main result
- Mass (kg)
- Final temperature (°C)
- Specific heat capacity (J/(kg·K))
- Temperature change (K)
- Heat (kWh)
- Carnot efficiency
- Maximum work from that heat (J)
- Best refrigerator COP
- Best heat-pump COP

## Formula

$$
Q = mc\Delta T,\qquad Q = mL,\qquad \eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H}
$$

## Worked examples

### Heat 1 kg of water from 20 °C to 100 °C

- Calculate: Temperature change
- Solve for: Heat
- Material: Water (liquid, 15 °C)
- Mass: 1 kg
- Initial temperature: 20 °C
- Final temperature: 100 °C
- **Heat: 334,880 J**
- **Heat: 0.093022 kWh**
- Checked against: Python 3.8 fractions: 1 × 4186 × 80 = 334880 J (OpenStax c_water = 4186)

### Cooling releases heat (negative Q)

- Calculate: Temperature change
- Solve for: Heat
- Material: Water (liquid, 15 °C)
- Mass: 1 kg
- Initial temperature: 80 °C
- Final temperature: 20 °C
- **Heat: -251,160 J**
- Checked against: Python 3.8: 1 × 4186 × (20 − 80)

### 9 kJ into 500 g of aluminium at 20 °C

- Calculate: Temperature change
- Solve for: Final temperature
- Material: Aluminium
- Heat added (negative if removed): 9 kJ
- Mass: 500 g
- Initial temperature: 20 °C
- **Final temperature: 40 °C**
- **Temperature change: 20 K**
- Checked against: Python 3.8 fractions: ΔT = 9000/(0.5 × 900) = 20 K

### Identify a metal: 3870 J warms 1 kg by 10 K

- Calculate: Temperature change
- Solve for: Specific heat
- Heat added (negative if removed): 3870 J
- Mass: 1 kg
- Initial temperature: 20 °C
- Final temperature: 30 °C
- **Specific heat capacity: 387 J/(kg·K)**
- Checked against: Python 3.8: 3870/(1 × 10) = 387 J/(kg·K), copper in OpenStax Table 1.3

### Melt 2 kg of ice

- Calculate: Phase change
- Mass: 2 kg
- Solve for: Heat
- Substance: Water
- Change: Melting / freezing
- **Heat: 668,000 J**
- Checked against: Python 3.8: 2 × 334 kJ/kg (OpenStax Table 1.4)

### Boil away 500 g of water

- Calculate: Phase change
- Mass: 0.5 kg
- Solve for: Heat
- Substance: Water
- Change: Boiling / condensing
- **Heat: 1,128,000 J**
- Checked against: Python 3.8: 0.5 × 2256 kJ/kg (OpenStax Table 1.4)

## Questions

### How much energy does it take to heat water?

4,186 J per kilogram per degree Celsius, water's specific heat capacity. Heating 1 litre (1 kg) from 20 °C to 100 °C takes 1 × 4,186 × 80 = 334,880 J, or 0.093 kWh. In US units that is about 1 BTU per pound per °F, which is how the BTU was originally defined.

### What is specific heat capacity?

The heat needed to raise 1 kg of a substance by 1 K (the same as 1 °C), in J/(kg·K). Water's is 4,186, among the highest of common substances, while copper's is 387 and lead's 128. The same 10 kJ warms 1 kg of water by 2.4 °C but 1 kg of copper by 25.8 °C, which is why water is used for cooling and heat storage.

### What is latent heat?

The energy absorbed or released during a phase change at constant temperature, Q = mL. For water the latent heat of fusion is 334 kJ/kg at 0 °C and of vaporisation 2,256 kJ/kg at 100 °C (OpenStax Table 1.4). Melting 2 kg of ice takes 668 kJ, enough to heat the same 2 kg of water by about 80 °C.

### What is the Carnot efficiency?

η = 1 − Tc/Th, the largest fraction of heat that any engine can turn into work between a hot reservoir at Th and a cold one at Tc, both in kelvin. Between 500 K and 300 K it is 40%; between boiling and freezing water, 26.8%. Real engines fall short of it: coal-fired power stations typically convert about 37% of their fuel's heat into electricity.

### What is the maximum COP of a heat pump?

COP = Th/(Th − Tc) with temperatures in kelvin, the Carnot limit on heat delivered per unit of work. Pumping heat from 0 °C outdoors into a 35 °C heating loop allows at most 308.15/35 ≈ 8.8. At −10 °C outside the limit drops to 6.8, and real machines stay well below it because of compressor and heat-exchanger losses.

### How accurate is the specific heat, latent heat and Carnot efficiency calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 10 worked examples whose answers come from independent sources; for example, “Heat 1 kg of water from 20 °C to 100 °C” is checked against Python 3.8 fractions: 1 × 4186 × 80 = 334880 J (OpenStax c_water = 4186).

### Where does the method come from?

OpenStax University Physics Volume 2, §1.5 Heat transfer, specific heat and calorimetry (Table 1.3); §1.6 Phase changes (Table 1.4); OpenStax University Physics Volume 2, §4.5 The Carnot cycle.

## Sources

- [OpenStax University Physics Volume 2, §1.5 Heat transfer, specific heat and calorimetry (Table 1.3); §1.6 Phase changes (Table 1.4)](https://openstax.org/books/university-physics-volume-2/pages/1-5-heat-transfer-specific-heat-and-calorimetry)
- [OpenStax University Physics Volume 2, §4.5 The Carnot cycle](https://openstax.org/books/university-physics-volume-2/pages/4-5-the-carnot-cycle)