Problem 54 - Chemistry An Atoms First Approach

Question

A person accidentally swallows a drop of liquid oxygen, \(\mathbf{O}_{2}(l)\) which has a density of \(1.149 \mathrm{g} / \mathrm{mL}\). Assuming the drop has a volume of \(0.050 \mathrm{mL},\) what volume of gas will be produced in the person's stomach at body temperature \(\left(37^{\circ} \mathrm{C}\right)\) and a pressure of \(1.0 \mathrm{atm}\)?

Step-by-Step Solution

Answer

After converting the temperature to Kelvin (310.15 K) and calculating the moles of oxygen (0.00179625 mol), we can use the Ideal Gas Law equation to find the volume of gas produced in the person's stomach: \(V = \frac{nRT}{P}\). Substituting the values, we get: \(V = \frac{(0.00179625 \,\text{mol}) * (0.0821\, \text{L·atm/mol·K}) * (310.15\, \text{K})}{(1.0\, \text{atm})} \approx 0.04571 \,\text{L}\).

Step 1: To convert the temperature from Celsius to Kelvin, we add 273.15 to the given temperature in Celsius: T(K) = T(°C) + 273.15 T(K) = 37°C + 273.15 T(K) = 310.15 K #Step 2: Calculate the mass of liquid oxygen#

We are given the density (\(\rho\)) of liquid oxygen and the volume (V) of the swallowed drop. We can use the formula for density to determine the mass (m) of liquid oxygen: \(\rho = \frac{m}{V}\) Rearranging the formula to find the mass, we get: \(m = \rho * V\) Substituting the values given, we get: m = (1.149 g/mL) * (0.050 mL) m = 0.05745 g #Step 3: Calculate the moles of oxygen#

Step 2: To find the moles (n) of oxygen, we will use the formula: n = m / M where M is the molar mass of O2 (32 g/mol). Substituting the values, we get: n = 0.05745 g / 32 g/mol n = 0.00179625 mol #Step 4: Use Ideal Gas Law Equation to find volume of gas produced#

We are given the pressure (P = 1.0 atm) and know the temperature already in Kelvin. The Ideal Gas Law equation is: PV = nRT Rearranging the equation to solve for V, we get: V = nRT/P Substituting the values, we get: V = (0.00179625 mol) * (0.0821 L·atm/mol·K) * (310.15 K) / (1.0 atm) V = 0.04571 L Therefore, the volume of gas produced in the person's stomach is approximately 0.04571 L.

Key Concepts

Liquid Oxygen

Liquid oxygen is simply oxygen that has been cooled to a liquid state. This occurs at extremely low temperatures, well below its boiling point of -183°C. At these temperatures, oxygen becomes a pale blue liquid and is useful for various industrial and medical applications.
It is important to note that when in liquid form, oxygen is denser than in its gaseous state. For example, the density of liquid oxygen is 1.149 g/mL. This means that a small volume of liquid oxygen can produce a large volume of gas when converted back.

• Liquid oxygen is cryogenic, meaning it requires extremely low temperatures to remain in liquid form.
• It is highly reactive due to its ability to support combustion.
• Because of its high density, small volumes can unleash large amounts of gas when warmed.

Molar Mass

Molar mass is an important concept in chemistry that refers to the mass of one mole of a given substance. For molecules, such as oxygen ext { f{O}}_2, the molar mass is calculated by adding the atomic masses of the atoms that form the molecule. In this case, the molar mass of ext { f{O}}_2 is 32 g/mol, since each oxygen atom has an atomic mass of about 16 g/mol.
Understanding molar mass is crucial for converting mass into moles, a fundamental process in chemical calculations. For instance, to find the moles of oxygen swallowed, you would divide the mass of liquid oxygen by its molar mass.

• Molar mass allows for conversions between grams and moles.
• For oxygen gas, ext { f{O}}_2, the molar mass is 32 g/mol.
• Molar mass is integral to the Ideal Gas Law for volume calculations.

Density Formula

The density formula is a simple yet powerful tool in science. It is defined as the mass of a substance per unit of volume and can be expressed as ext { f{Density}} ( ho) = rac{ ext{mass}}{ ext{volume}}. This formula is useful for calculating the mass of a material when its volume and density are known or vice versa. The exercise uses this formula to determine how much liquid oxygen was swallowed in terms of its mass. The density is set, and by knowing the volume, you find mass simply by multiplying the two.

• Density is utilized to relate mass and volume.
• It helps identify substances and their property characteristics.
• In this case, density allows calculation of the mass of liquid oxygen swallowed.

Temperature Conversion

When working with temperatures in physical sciences, conversions between Celsius and Kelvin are often required. The Kelvin scale is preferred in calculations involving gas laws because it starts at absolute zero, the theoretically lowest temperature possible. To convert from Celsius to Kelvin, the formula used is ext { f{T(K)}} = ext { f{T(}^ ext { f{o}} ext{C)}} + 273.15. This was used in the exercise to convert body temperature from 37°C to 310.15 K for use in the Ideal Gas Law equation. The conversion is crucial as gas laws require temperatures in absolute terms to accurately predict behavior.

• Kelvin is the SI unit for temperature and needed for gas law equations.
• Conversion ensures calculations align with scientific standards.
• A simple addition of 273.15 is all that's needed between Celsius and Kelvin.