Question

The space shuttle uses liquid hydrogen as its fuel. The external fuel tank used during takeoff carries \(227,641 \mathrm{lb}\) of hydrogen with a volume of 385,265 gallons. Calculate the density of liquid hydrogen in units of \(\mathrm{lb} / \mathrm{gal}\) and \(\mathrm{g} / \mathrm{mL}\). (Express your answer to three significant figures.) What is the specific gravity of liquid hydrogen?

Short answer

The density of liquid hydrogen is approximately 0.590 lb/gal or 0.0706 g/mL. The specific gravity, rounded to three significant figures, is 0.0706.

Step-by-step solution

Calculate the density in lb/gal

To find the density of liquid hydrogen in lb/gal, divide the mass of the hydrogen (in pounds) by its volume (in gallons). Density formula: \( \text{density} = \frac{\text{mass}}{\text{volume}} \).

Convert the density to g/mL

First, convert the density from lb/gal to lb/L by multiplying it by 0.264172 (since 1 gallon is approximately 3.78541 liters), then convert pounds to grams by multiplying by 453.592 (since 1 pound is approximately 453.592 grams), and finally divide by 1000 to convert from L to mL.

Round the converted density to three significant figures

Round the calculated density from the previous step to three significant figures to express your final answer to the required precision.

Calculate the specific gravity of liquid hydrogen

To find the specific gravity, divide the density of liquid hydrogen by the density of water in the same units. The density of water is approximately 1 g/mL or 8.3454 lb/gal. Use the density in g/mL for the calculation to find the specific gravity.

Round the specific gravity to three significant figures

Round the specific gravity calculated in the previous step to three significant figures.

Key concepts

Liquid Hydrogen

Liquid hydrogen is a common fuel used in space exploration, particularly because of its high energy per unit mass compared to other fuels. In its liquid form, hydrogen is stored at very low temperatures, below its boiling point of -423°F (-253°C). The space shuttle, as mentioned in the provided example, uses liquid hydrogen due to its efficiency and potent energy release upon combustion with oxygen. Understanding the properties of liquid hydrogen, including its density, is crucial for applications in aerospace as it impacts the design and performance of the fuel tank and the overall vehicle.

Density is a fundamental physical property defined as mass per unit volume. It's essential for engineers to know the density of liquid hydrogen to ensure that the fuel tanks can store an adequate amount while still conforming to size and weight constraints of the spacecraft. The challenge in dealing with liquid hydrogen lies in its extremely low storage temperature, which requires specialized insulated tanks to prevent evaporation and maintain its liquid state.

Specific Gravity

Specific gravity is a dimensionless quantity that represents the ratio of a substance's density to the density of a reference substance, typically water at 4°C (39°F) where it has the density of 1 g/mL. For any substance, the specific gravity is calculated with the formula: \[ \text{specific gravity} = \frac{\text{density of substance}}{\text{density of water}} \].

In practical terms, the specific gravity can give us a quick comparison to see if the substance is lighter or heavier than water. For liquid hydrogen, the specific gravity is much less than 1, signifying that it is much lighter than water and will float if placed in water. This property is beneficial in the context of the space shuttle; lighter fuel contributes to a more efficient launch, where every ounce of weight matters. Knowing the specific gravity of the fuel also helps in controlling the buoyancy of components in various engineering applications.

Conversion of Units

The conversion of units is a critical skill in science and engineering, as measurements can be taken or required in different units depending on the context or standards. The problem provided requires us to convert pounds per gallon (lb/gal) to grams per milliliter (g/mL), which is a multi-step conversion process. It involves converting volumes from gallons to liters, then to milliliters, and mass from pounds to grams.

For volume conversion, we use the fact that 1 gallon equals approximately 3.78541 liters, and there are 1000 milliliters in a liter. For mass conversion, 1 pound is equivalent to 453.592 grams. It’s important to use precise factors for conversion to minimize errors, and these factors are derived from definitions and standards of measurements agreed upon internationally. Keeping track of units throughout this process helps avoid mistakes and ensures that the final units reflect the desired quantity.

Significant Figures

Significant figures are a way of expressing precision in measurements and calculations. They represent all the valid digits in a number, starting from the first non-zero digit. In calculations like density and specific gravity, it's crucial to express the answer with the correct number of significant figures to reflect the precision of the measurement. This often involves rounding off after a calculation is completed.

In the space shuttle problem, we are requested to provide answers to three significant figures. This limitation not only guides us in rounding off our final answers but also reminds us to consider the precision of measuring tools and processes used. Not adhering to proper significant figures can result in misleading precision, which in scientific and engineering practices can lead to inaccuracies with potentially critical consequences. When completing the calculations, pay attention to these figures at each mathematical step to ensure the final result is both accurate and appropriately precise.