For which of the following parameters, the structural isomers C2H5OH and CH3OCH3 would be expected t

Q: Why doesn't dimethyl ether form hydrogen bonds?

For hydrogen bonding to occur, hydrogen must be directly attached to a highly electronegative atom like F, O, or N. In dimethyl ether, all hydrogens are attached to carbon.

Q: Doesn't higher boiling point mean higher density?

Not for gases! In the liquid phase, intermolecular forces affect density. But for an *ideal gas*, we assume no intermolecular forces exist, so density depends only on molar mass, pressure, and temperature.

Q: How did we get $d = PM/RT$?

Start with $PV = nRT$. Substitute $n = m/M$ (mass/molar mass) to get $PV = (m/M)RT$. Rearrange to $P = (m/V)(RT/M)$. Since density $d = m/V$, we get $P = dRT/M$, or $d = PM/RT$.

Q: What if the question didn't say 'assume ideal behaviour'?

If they were real gases, the slight differences in their intermolecular forces (van der Waals vs H-bonding) would cause them to deviate differently from ideal behaviour, meaning their gaseous densities wouldn't be exactly identical.

Q: Why is the heat of vaporization different?

Heat of vaporization is the energy required to overcome intermolecular forces and turn a liquid into a gas. Since ethanol has strong H-bonds, it requires much more energy to vaporize than ether.

Question

For which of the following parameters, the structural isomers $C_2H_5OH$ and $CH_3OCH_3$ would be expected to have the same values? (Assume ideal behaviour)

Options:

(A) Heat of vaporisation

(B) Vapour pressure at the same temperature

(C) Boiling points

(D) Gaseous densities at the same temperature and pressure

Sigma Percentile · Accepted Answer

### The Tale of Two Isomers

Imagine you are holding two flasks. One contains ethanol, the familiar alcohol, and the other contains dimethyl ether, a highly volatile gas at room temperature. Both of these compounds are structural isomers. This means they share the exact same molecular formula, $C_2H_6O$, but their atoms are arranged differently.

In ethanol, the oxygen atom is bonded to a hydrogen atom, forming an $-OH$ group. In dimethyl ether, the oxygen atom is sandwiched between two methyl groups ($-CH_3$). This seemingly small difference in architecture leads to a massive difference in their physical properties.

### The Power of the Hydrogen Bond

Because ethanol has an $-OH$ group, its molecules can engage in **intermolecular hydrogen bonding**. This is a strong dipole-dipole interaction where the slightly positive hydrogen of one molecule is attracted to the lone pairs on the oxygen of a neighboring molecule.

Dimethyl ether, lacking an $-OH$ group, can only rely on much weaker dipole-dipole forces and London dispersion forces.

What does this mean in the real world? It means ethanol molecules hold onto each other tightly. To pull them apart and turn the liquid into a gas, you need to supply a lot of energy. Therefore, ethanol has a **higher heat of vaporization** and a **higher boiling point** than dimethyl ether. Furthermore, because the molecules are reluctant to escape the liquid phase, ethanol has a **lower vapour pressure** at any given temperature.

So, if we look at our options, heat of vaporization, vapour pressure, and boiling point will all be drastically different for these two isomers.

### The Ideal Gas Equalizer

But what happens when both substances are already in the gaseous state, and we assume they behave as **ideal gases**?

The problem explicitly tells us to assume ideal behaviour. In the realm of ideal gases, intermolecular forces are completely ignored. The identity of the gas no longer matters; only the number of particles, the volume, the temperature, and the pressure matter.

Let's bring in the master equation of state:
$$PV = nRT$$

We know that the number of moles ($n$) is equal to the given mass ($m$) divided by the molar mass ($M$). Substituting this in, we get:
$$PV = \frac{m}{M}RT$$

Now, let's rearrange this to solve for density ($d$), which is mass divided by volume ($m/V$):
$$P = \frac{m}{V} \frac{RT}{M}$$
$$P = d \frac{RT}{M}$$
$$d = \frac{PM}{RT}$$

### The Final Verdict

Look closely at this final equation. The density of an ideal gas depends on the pressure ($P$), the temperature ($T$), the universal gas constant ($R$), and the **molar mass** ($M$).

The question asks us to compare their gaseous densities at the **same temperature and pressure**. Since $P$, $R$, and $T$ are constant for both gases, the density is directly proportional to the molar mass.

Because ethanol and dimethyl ether are structural isomers, they have the exact same molecular formula ($C_2H_6O$) and, consequently, the **exact same molar mass**.

Therefore, under ideal conditions, their gaseous densities will be perfectly identical. The correct answer is **Option (d)**.

Animated Solution for Chemistry - States of Matter: For which of the following parameters, the structural isomers $C_{2} H_{5} O H$ and $C H_{3} O C H_{3}$ would be expected to have the same values? (Assume ideal behaviour)

Select Answer:

Visualized Solution

$C_{2} H_{6} O$

$H - bonds$

$B P \propto Forces$

$P V = n R T$

$d = \frac{P M}{R T}$

The Sigma Insight: Gaseous State

The Tale of Two Isomers

The Power of the Hydrogen Bond

The Ideal Gas Equalizer

The Final Verdict

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