Problem 27 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

\(10.30 \mathrm{mg}\) of \(\mathrm{O}_{2}\) is dissolved into a liter of sea water of density \(1.03\) \(\mathrm{g} / \mathrm{mL}\). The concentration of \(\mathrm{O}_{2}\) in ppm is Main Jan. 09. 2020 (II)

Step-by-Step Solution

Answer

The concentration of \(\mathrm{O}_{2}\) in sea water is 10.01 ppm.

Step 1: Convert Milligrams to Grams

First, we must convert the mass of \(\mathrm{O}_{2}\) from milligrams to grams for consistency in units. Since there are 1000 milligrams in a gram, we have: \ 10.30 \, \mathrm{mg} = \frac{10.30}{1000} \, \mathrm{g} = 0.0103 \, \mathrm{g}\.

Step 2: Calculate the Density of Sea Water in Grams per Liter

Given that the density of sea water is \(1.03 \, \mathrm{g/mL}\), we convert this to \(\mathrm{g/L}\). Since there are 1000 mL in a liter, the density of sea water is: \ 1.03 \, \mathrm{g/mL} \times 1000 \, \mathrm{mL/L} = 1030 \, \mathrm{g/L}\.

Step 3: Calculate the Fraction of \\(\mathrm{O}_{2}\\) by Mass

To find the fraction by mass of \(\mathrm{O}_{2}\), divide the mass of \(\mathrm{O}_{2}\) by the mass of one liter of sea water: \ \frac{0.0103 \, \mathrm{g}}{1030 \, \mathrm{g}} = 1.00097 \times 10^{-5}\.

Step 4: Convert the Fraction to Parts Per Million (ppm)

Parts per million is defined as parts of solute per million parts of solution by mass. Thus, multiply the fraction of \(\mathrm{O}_{2}\) by one million: \ 1.00097 \times 10^{-5} \times 10^6 = 10.0097 \, \text{ppm}\. Round this to four significant figures to match the given data: \ \text{10.01} \, \text{ppm}\.

Key Concepts

Oxygen Concentration

Oxygen concentration in water, such as seawater, is an essential metric to understand. For marine life to thrive, the right levels of dissolved oxygen must be maintained. In this particular problem, we're interested in expressing the concentration of oxygen in a very specific unit called parts per million (ppm).

In chemistry, ppm is used to measure the small concentrations of solutes in a solution. It refers to the number of parts of a substance in one million parts of the total solution. Calculating the ppm involves converting the mass of the solute (oxygen in our case) into a comparable unit with the density of the solution, eventually multiplying by one million to find the desired concentration value. The challenge lies in ensuring all units are consistent before performing these conversions.

Density Conversion

Density is the mass of a substance per unit volume. For scenarios like this problem where we're dealing with solutions and small concentrations, understanding the density in correct units is crucial. Here, we start with the density of seawater given in grams per milliliter (g/mL).

To efficiently utilize this information in calculations with larger units such as liters, conversion is necessary. Understanding that there are 1000 milliliters in a liter allows you to switch from g/mL to g/L by multiplying the density by 1000. Similarly, this fundamental conversion step ensures that calculations involving mass of the solute and the solution volume match in terms of units, easing further calculations.

Mass Fraction

A mass fraction describes how much of one substance is in a mixture, expressed as a fraction of the total mass. It is an important concept, especially in diluted solutions, to accurately determine the concentration of a specific component.

In the context of this problem, the mass fraction of oxygen is found by dividing the mass of oxygen by the total mass of seawater per liter. While it seems simple, it effectively scales down the larger mass of the solvent (seawater) against the relatively diminutive mass of the oxygen solute, creating a dimensionless number that provides insight into the oxygen's relative abundance in the mixture. This mass fraction is the stepping stone to converting the values into ppm, a more practical unit for parts-per notation.

Unit Conversion

Unit conversion is a cornerstone in solving chemistry problems involving different measurements. It ensures that you are working with comparable units, thus maintaining the integrity of your calculation across various dimensions.

This involves:

• Converting milligrams to grams for ease of mass comparisons.
• Switching milliliters to liters to express density for volumetric computations.
• Scaling fractions by a factor (such as one million) to shift into parts per notation (ppm).

Accuracy in unit conversion prevents errors at each solution step. It ties together concepts like mass fraction and density conversion cohesively, ultimately leading to the correct determination of concentration expressed in ppm.