The isothermal bulk modulus of a gas at atmospheric pressure is

To find the isothermal bulk modulus of a gas at atmospheric pressure, we begin by defining the isothermal bulk modulus.

The bulk modulus ($B$) of a substance measures its resistance to uniform compression and is defined as the ratio of the infinitesimal pressure increase to the resulting relative decrease of the volume:
$B=-V\frac{dP}{dV}$
where $P$ is pressure and $V$ is volume.

For an ideal gas undergoing an isothermal process (a process occurring at a constant temperature $T$), the system obeys Boyle's Law:
$PV=\text{constant}$

To find $\frac{dP}{dV}$, we differentiate both sides of the equation $PV=\text{constant}$ with respect to $V$ using the product rule:
$P\frac{dV}{dV}+V\frac{dP}{dV}=0$
$P+V\frac{dP}{dV}=0$
Rearranging the terms, we get:
$-V\frac{dP}{dV}=P$

Substituting this back into the definition of the bulk modulus, we find that the isothermal bulk modulus ($B_{\text{isothermal}}$) of an ideal gas is equal to its pressure:
$B_{\text{isothermal}}=P$

We are given that the gas is at atmospheric pressure. The standard value of atmospheric pressure at sea level is:
$P=1\text{atm}=1.013\times {10}^5{\text{N / m}}^2$

Therefore, the isothermal bulk modulus of the gas is:
$B_{\text{isothermal}}=1.013\times {10}^5{\text{N / m}}^2$

The correct option is 1.013 x 10⁵ N / m².