One mole of a monoatomic gas is mixed with three moles of a diatomic gas. The molecular specific heat of mixture at constant volume is \(\frac{\alpha^2}{4}\) RJ/mol k; then the value of α will be _______.
(Assume that the given diatomic gas has no vibrational mode).
(Assume that the given diatomic gas has no vibrational mode).
Correct Answer: 3
Approach Solution - 1
To find the value of \( \alpha \), we analyze the specific heat at constant volume \( C_v \) for a mixture of monoatomic and diatomic gases.
For a monoatomic gas, \( C_{v,\text{mono}} = \frac{3}{2}R \).
For a diatomic gas without vibrational mode, \( C_{v,\text{di}} = \frac{5}{2}R \).
The mixture contains 1 mole of monoatomic and 3 moles of diatomic gases. The total specific heat is calculated as:
\[ C_{v,\text{mixture}} = \frac{1 \cdot C_{v,\text{mono}} + 3 \cdot C_{v,\text{di}}}{1 + 3} \]
Inserting the given values:
\[ C_{v,\text{mixture}} = \frac{1 \cdot \frac{3}{2}R + 3 \cdot \frac{5}{2}R}{4} = \frac{\frac{3}{2}R + \frac{15}{2}R}{4} = \frac{18}{8}R = \frac{9}{4}R \]
According to the problem, \( C_{v,\text{mixture}} = \frac{\alpha^2}{4}R \).
Equating both expressions for specific heat, we have:
\[ \frac{9}{4}R = \frac{\alpha^2}{4}R \]
Cancelling \( R \) and the denominator \( 4 \), we get:
\[ 9 = \alpha^2 \]
Taking the square root gives \( \alpha = 3 \).
We confirm \( 3 \) lies within the given range of \([3,3]\).
Thus, the value of \( \alpha \) is 3.
Approach Solution -2
\(C_V = \frac{f}{2}R\)
Total degree of freedoms
\(= 1 × 3 + 3 × 5 = 18\)
\(\frac{\alpha^2}{4} = \frac{18}{2n} = \frac{18}{2 \times 4}\)
\(⇒ α^2 = 9\)
\(α = 3\)
So, the answer is 3.
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Concepts Used:
Mole Concept
In the field of chemistry, a mole is defined as the amount of a substance that contains exactly 6.02214076 * 1023 ‘elementary entities’ of the given substance.
The number 6.02214076*1023 is popularly known as the Avogadro constant and is often denoted by the symbol ‘NA’. The elementary entities that can be represented in moles can be atoms, molecules, monoatomic/polyatomic ions, and other particles (such as electrons).
For example, one mole of a pure carbon-12 (12C) sample will have a mass of exactly 12 grams and will contain 6.02214076*1023 (NA) number of 12C atoms. The number of moles of a substance in a given pure sample can be represented by the following formula:
n = N/NA
Where n is the number of moles of the substance (or elementary entity), N is the total number of elementary entities in the sample, and NA is the Avogadro constant.
The word “mole” was introduced around the year 1896 by the German chemist Wilhelm Ostwald, who derived the term from the Latin word moles meaning a ‘heap’ or ‘pile.
The mole concept refers to the atomic mass of a mole that is measured in grams. The gram atomic mass of an element is known as a mole. The mole concept combines the mass of a single atom or molecule in a.m.u. to the mass of a large group of comparable molecules in grams. Atomic mass is the mass of a single atom, whereas molecular mass is the mass of a group of atoms.
The Formula of Mole Concept
The number of units that make up a mole has been calculated to be 6.022 ×10²³. The fundamental constant is also known as Avogadro's number (NA) or Avogadro constant. This constant is appropriately represented in chemistry using an explicit unit termed per mole.
Number of Moles = (Mass of the Sample)/(Molar Mass)
Read More: Mole Fraction