Question

Volume of a gas at NTP is \(1.12 \times 10^{-7} \mathrm{cc}\). The number of molecules in it is: (a) \(3.01 \times 10^{12}\) (b) \(3.01 \times 10^{18}\) (c) \(3.01 \times 10^{24}\) (d) \(3.01 \times 10^{30}\)

Short answer

The number of molecules is \(3.01 \times 10^{12}\), option (a).

Step-by-step solution

Understanding NTP Conditions

Normal Temperature and Pressure (NTP) conditions refer to a temperature of 273 K and a pressure of 1 atm. At NTP, 1 mole of an ideal gas occupies 22.4 L.

Convert Volume from cc to Liters

The volume given is in cubic centimeters (cc). First, convert this to liters, since 1 L = 1000 cc. So, the volume in liters is: \[ V = \frac{1.12 \times 10^{-7}}{1000} = 1.12 \times 10^{-10} \text{ L} \]

Calculate Moles of Gas

Using the fact that 1 mole of gas occupies 22.4 L at NTP, calculate the number of moles from the volume:\[ \text{moles} = \frac{1.12 \times 10^{-10}}{22.4} \approx 5 \times 10^{-12} \text{ moles} \]

Use Avogadro's Number to Find Molecules

Avogadro's number tells us that 1 mole of any substance contains approximately \(6.022 \times 10^{23}\) molecules. Use this to find the number of molecules:\[ \text{number of molecules} = 5 \times 10^{-12} \times 6.022 \times 10^{23} \approx 3.01 \times 10^{12} \]

Final Answer Selection

Compare the calculated number of molecules, \(3.01 \times 10^{12}\), with the provided options. The correct choice is (a) \(3.01 \times 10^{12}\).

Key concepts

NTP conditions

Normal Temperature and Pressure (NTP) conditions are a specific reference point in chemistry used to standardize the description of the physical properties of gases. Under NTP, the temperature is usually set at 273 Kelvin (K), which is equivalent to 0 degrees Celsius, and the pressure is set at 1 atmosphere (atm). These conditions are important because they allow scientists to predict and compare the behavior of gases under standardized conditions. At NTP, 1 mole of an ideal gas is known to occupy a volume of 22.4 liters (L). This equivalence is crucial in stoichiometry calculations involving gases. It's worth noting that the NTP conditions can slightly differ from the more frequently mentioned Standard Temperature and Pressure (STP), which was traditionally 0°C and 1 atm before being updated to 273.15 K and 1 atm by IUPAC.

Volume conversion

Converting volume units is an essential skill, especially when working with gases in different scientific contexts. In this exercise, the volume is initially given in cubic centimeters (cc), which is a common unit for small volumes. However, in most gas calculations, it's standard to use liters (L) as the unit of volume.
To convert from cc to liters, remember that:

• 1 liter = 1000 cubic centimeters

Thus, to convert to liters, you divide the volume in cubic centimeters by 1000. For example, a gas volume of \(1.12 \times 10^{-7}\) cc converts to liters as follows:

• \( V = \frac{1.12 \times 10^{-7}}{1000} = 1.12 \times 10^{-10} \text{ L} \)

This conversion helps align the volume unit with the conditions at NTP, where 1 mole of gas occupies 22.4 L.

Avogadro's Number

Avogadro's number is a fundamental concept in chemistry, representing the number of constituents, such as atoms or molecules, per mole of a substance. Its value is approximately \(6.022 \times 10^{23}\). This number allows chemists to relate the mass of a substance to the number of particles or entities it contains.
Avogadro's number is crucial when dealing with gases, especially at NTP. Once the number of moles present in a known volume of gas is calculated, Avogadro’s number helps to determine the exact number of molecules in that gas. For example, if you find that you have \(5 \times 10^{-12}\) moles of a gas, as calculated in step 3, by multiplying this number by Avogadro's number, you accurately count the number of molecules:

• \( \text{number of molecules} = 5 \times 10^{-12} \times 6.022 \times 10^{23} \approx 3.01 \times 10^{12} \)

Avogadro's number bridges the macroscopic and microscopic worlds, facilitating precise conversions from moles to particles.

Mole calculations

Moles are a fundamental unit in chemistry used to express amounts of a chemical substance. A mole is defined as the amount of any chemical substance that contains as many elementary entities, such as atoms, molecules, ions, etc., as there are in 12 grams of pure carbon-12. Using the mole concept, chemists can easily convert between the mass of a substance and the number of particles it contains.
At NTP, the relationship between the volume of gas and the number of moles is particularly clear: 1 mole occupies 22.4 liters. This property allows scientists to precisely determine the number of moles from a given volume of gas. For example, if you're given a volume of \(1.12 \times 10^{-10}\) liters of gas, you can find out how many moles this equates to at NTP:

• \( \text{moles} = \frac{1.12 \times 10^{-10}}{22.4} \approx 5 \times 10^{-12} \text{moles} \)

This calculation is key for proceeding to determine the molecule count in any given sample using Avogadro's number.