Question

A gas company in Massachusetts charges \(\$ 1.30\) for \(15.0 \mathrm{ft}^{3}\) of natural gas. (a) Convert this rate to dollars per liter of gas. (b) If it takes \(0.304 \mathrm{ft}^{3}\) of gas to boil a liter of water, starting at room temperature \(\left(25^{\circ} \mathrm{C}\right)\), how much would it cost to boil a \(2.1-\mathrm{L}\) kettle of water?

Short answer

The cost per liter of gas is \(0.244\) dollars per liter and it would cost \(0.516\) dollars to boil a \(2.1L\) kettle of water.

Step-by-step solution

Conversion from cubic feet to liters

To convert the rate from cubic feet to liters, we need to understand the relation between the two units. 1 cubic foot (ft³) is approximately equal to 28.3168 liters (L). So, \(1.30 ÷ 15.0\) gives us the cost per cubic foot of gas. Then we multiply it by \(28.3168\) to find the cost per liter of gas: \(\(1.3 ÷ 15.0 \cdot 28.3168\) \).

Calculating the cost to boil one liter of water

We are given that it takes \(0.304 \mathrm{ft}^3\) of gas to boil one liter of water. So, we need to find how much 1 cubic foot (0.304) of gas would cost. We do this by multiplying the cost per liter of gas (found in the previous step) by \(0.304\).

Calculating the cost to boil a 2.1-L kettle of water

Finally, after that conversion, we need to calculate the cost to boil a \(2.1L\) kettle of water. We do this by multiplying the cost to boil one liter of water (calculated in the previous step) by \(2.1\) .

Key concepts

Unit Conversion

Understanding unit conversion is pivotal in chemistry, especially when it comes to practical applications like calculating costs. The process involves changing from one unit of measurement to another without altering the actual amount of substance. For example, in the provided exercise, we convert gas volume from cubic feet to liters — two different units that measure volume.

We use the conversion factor that 1 cubic foot is approximately equal to 28.3168 liters. With this factor, we can scale the volume cost ratio accordingly. The step-by-step solution illustrates this by dividing the cost per cubic foot by the volume in cubic feet and then multiplying by the conversion factor to get the price per liter. This conversion is essential because liters are a more common unit of measure for volumes of liquids in many parts of the world, including the scientific community.

Stoichiometry

Stoichiometry is the section of chemistry that involves calculating the proportions and quantities of reactants and products in chemical reactions. It's like a recipe that tells you how much of each ingredient you need for the desired outcome.

In the context of our gas conversion problem, stoichiometry is used to determine the volume of gas required to boil water. It relates the volume of gas used to the volume of water boiled. The problem provides the volume of gas necessary for boiling a liter of water, and this proportion can be used to find the volume required for any amount of water — in our case, a 2.1-liter kettle. Once we know this amount, we can apply the cost per liter of gas to find the total cost to boil the 2.1 liters of water. This concept is key in scaling up reactions and processes in a cost-effective manner.

Energy Cost Calculations

Energy cost calculations are integral in assessing the economic aspect of chemical processes. It involves computing the monetary cost of energy consumed during a process. This information can lead to more informed decisions regarding energy use and efficiency.

In our exercise, the solution shows how to find the cost to boil water based on the energy source's cost—natural gas, in this case. After determining the cost per liter of gas from the unit conversion, we multiply this cost by the amount of gas needed to boil one liter of water. Finally, we scale this up to a 2.1-liter kettle. Such calculations are not only crucial for household budgeting but also play a significant role in industrial processes, where optimizing energy usage can lead to substantial cost savings.