Problem 42 - Chemistry: An Atoms-Focused Approach

Question

Which has more heat capacity: one liter of water or one cubic meter of water? Which has more molar heat capacity?

Step-by-Step Solution

Answer

Question: Compare the heat capacity and molar heat capacity of one liter of water and one cubic meter of water. Answer: One cubic meter of water has a higher heat capacity than one liter of water. However, the molar heat capacity is the same for both one liter and one cubic meter of water, at 75.24 J/(mol°C).

Step 1: Calculate the mass of water in both cases

One liter of water is equivalent to 1,000 cubic centimeters (1 L = 1,000 cm³). Since water has a density of approximately 1 gram per cubic centimeter, one liter of water has a mass of about 1,000 grams or 1 kilogram. One cubic meter of water is equivalent to 1,000,000 cubic centimeters (1 m³ = 1,000,000 cm³). Therefore, one cubic meter of water has a mass of about 1,000,000 grams or 1,000 kilograms.

Step 2: Calculate the heat capacities

To compute the heat capacities, we'll use the formula: Heat capacity = Mass x Specific heat capacity. The specific heat capacity of water is around 4.18 J/(g°C). Heat capacity of one liter of water = (1,000 g) x (4.18 J/(g°C)) = 4,180 J/°C Heat capacity of one cubic meter of water = (1,000,000 g) x (4.18 J/(g°C)) = 4,180,000 J/°C

Step 3: Find the number of moles of water in both cases

We can find the number of moles using the formula: Number of moles = Mass / Molar mass. The molar mass of water is approximately 18 g/mol. Number of moles of water in one liter: (1,000 g) / (18 g/mol) ≈ 55.56 moles Number of moles of water in one cubic meter: (1,000,000 g) / (18 g/mol) ≈ 55,556 moles

Step 4: Calculate the molar heat capacities

Now, we can compute the molar heat capacities using the formula: Molar heat capacity = Heat capacity / Number of moles. Molar heat capacity of one liter of water: 4,180 J/°C / 55.56 moles ≈ 75.24 J/(mol°C) Molar heat capacity of one cubic meter of water: 4,180,000 J/°C / 55,556 moles ≈ 75.24 J/(mol°C) Now comparing the results:

Step 5: Compare the heat capacities and molar heat capacities

Heat capacity: One liter of water: 4,180 J/°C One cubic meter of water: 4,180,000 J/°C Molar heat capacity: One liter of water: 75.24 J/(mol°C) One cubic meter of water: 75.24 J/(mol°C) In conclusion, one cubic meter of water has more heat capacity than one liter of water. However, both one liter and one cubic meter of water have the same molar heat capacity.

Key Concepts

Specific Heat Capacity

When studying the concept of specific heat capacity, it's crucial to understand that it represents the amount of heat energy required to raise the temperature of one gram of a substance by one degree Celsius. For water, a common substance, the specific heat capacity is approximately 4.18 J/(g°C). This value is relatively high compared to many other substances, making water an excellent heat reservoir.

In practical terms, this means that water can absorb a lot of heat without significantly changing in temperature. This is why environments near large bodies of water tend to have milder climates, as water can absorb and release large amounts of heat without a dramatic change in temperature.

To calculate the heat capacity of an object, you can use the equation:

• Heat capacity = Mass x Specific heat capacity

For example, with water's specific heat capacity, if you have 1 gram of water, it requires approximately 4.18 joules of energy to increase its temperature by one degree Celsius. This idea is fundamental in understanding how different substances react to energy changes.

Molar Heat Capacity

Molar heat capacity is another essential concept in thermodynamics, and it refers to the heat capacity of one mole of a substance. This concept is particularly useful when dealing with chemical reactions and processes where the number of moles is an important factor.

The molar heat capacity allows us to understand how much heat is required to change the temperature of a mole of substance by one degree Celsius. It's calculated using the formula:

• Molar heat capacity = Heat capacity / Number of moles

In our exercise example, the molar heat capacity of water is approximately 75.24 J/(mol°C), regardless of the volume of water. This is because the molar heat capacity is a property based on the substance itself, not its amount or volume.

Therefore, understanding molar heat capacity is crucial in chemistry, as it allows for the prediction of how substances in a reaction will behave when subjected to temperature changes.

Density of Water

The density of water is a fundamental property that significantly impacts its behavior in various physical scenarios. Density is defined as mass per unit volume, and for water, it is roughly 1 gram per cubic centimeter (or equivalently, 1,000 kilograms per cubic meter).

This means that for water, 1 liter (1,000 cubic centimeters) will have a mass of about 1 kilogram, and 1 cubic meter will have a mass of around 1,000 kilograms. This property allows us to determine the mass of water given its volume, which is essential when calculating heat capacity or conducting experiments that require precise measurements.

Density plays a vital role in many processes, including the buoyancy of objects and the layering of lakes and oceans. Due to its density, water can also absorb a lot of heat energy, contributing to its high specific heat capacity.

Having a clear understanding of the density of water is important in both academic studies and practical applications, as it often serves as a baseline for measuring and comparing other substances.