(Take $R_0 = 1.2 \times 10^{-15} \text{ m}$, $4\pi = 12.56$)
Nuclear Density Calculation Steps
To find the approximate mass number (A) of the nucleus, we use the relationship between nuclear density ($\rho$), the mass of the nucleus ($m$), and the nuclear radius ($R$).
Nuclear Density Formula
Nuclear density is defined as mass per unit volume:
$ \rho = \frac{m}{V} $The volume ($V$) of a nucleus is approximated as a sphere with radius $R$:
$ V = \frac{4}{3}\pi R^3 $The nuclear radius ($R$) is related to the mass number ($A$) by the empirical formula:
$ R = R_0 A^{1/3} $Where $R_0$ is a constant ($1.2 \times 10^{-15} \text{ m}$).
Combining Formulas
Substituting the expression for $R$ into the volume formula:
$ V = \frac{4}{3}\pi (R_0 A^{1/3})^3 = \frac{4}{3}\pi R_0^3 A $Now, substitute this volume into the density formula:
$ \rho = \frac{m}{\frac{4}{3}\pi R_0^3 A} $Calculating Mass Number A
Rearrange the formula to solve for the mass number ($A$):
$ A = \frac{m}{\rho \frac{4}{3}\pi R_0^3} $Given values:
- Mass ($m$) = $19.926 \times 10^{-27} \text{ kg}$
- Nuclear density ($\rho$) = $2.29 \times 10^{17} \text{ kg/m}^3$
- $R_0 = 1.2 \times 10^{-15} \text{ m}$
- $4\pi \approx 12.56$
Calculate the term $\frac{4}{3}\pi R_0^3$:
$ \frac{4}{3}\pi R_0^3 \approx \frac{1}{3} \times (12.56) \times (1.2 \times 10^{-15} \text{ m})^3 $ $ \frac{4}{3}\pi R_0^3 \approx \frac{12.56}{3} \times (1.728 \times 10^{-45} \text{ m}^3) $ $ \frac{4}{3}\pi R_0^3 \approx 4.1867 \times 1.728 \times 10^{-45} \text{ m}^3 \approx 7.235 \times 10^{-45} \text{ m}^3 $Now substitute all values into the formula for $A$:
$ A = \frac{19.926 \times 10^{-27} \text{ kg}}{(2.29 \times 10^{17} \text{ kg/m}^3) \times (7.235 \times 10^{-45} \text{ m}^3)} $ $ A = \frac{19.926 \times 10^{-27}}{2.29 \times 7.235 \times 10^{17 - 45}} $ $ A = \frac{19.926 \times 10^{-27}}{16.568 \times 10^{-28}} $ $ A = \frac{19.926}{1.6568} \approx 12.026 $Conclusion
The calculated mass number $A$ is approximately 12.026. Rounding to the nearest integer gives $A = 12$. This corresponds to Option 1.
Similar Questions
Match List I with List II :
List I List II A. $E = h\nu$ I. de Broglie wavelength B. Interference II. Particle nature of light C. $\lambda = h/p$ III. Wave nature of light D. Compton effect IV. Energy of photon
Choose the correct answer from the options given below :In the first excited state of hydrogen atom, the energy of its electron is $-3.4 \text{ eV}$. The radial distance of the electron from the hydrogen nucleus in this case is approximately :
(Take $1 \text{ eV} = 1.6 \times 10^{-19} \text{ J}, \text{ e} = 1.6 \times 10^{-19} \text{ C}$ and $\frac{1}{4\pi\varepsilon_0} = 9 \times 10^9 \text{ N m}^2/\text{C}^2$)Four statements are given (A is mass number) :
A. The volume of a nucleus is proportional to $A^{1/3}$.
B. The volume of a nucleus is proportional to A.
C. The difference in mass of an atom and its nucleus is called the mass defect.
D. The difference in mass of a nucleus and its constituent nucleons is called the mass defect.
Choose the correct answer from the options given below :- A ray of light with wavelength $\lambda$ is incident on three different photo-electric cells namely 1, 2 and 3. The threshold wavelength of these photo-electric cells are $\lambda_1$, $\lambda_2$, and $\lambda_3$, respectively and the magnitude of stopping potentials of these cells are $V_1$, $V_2$ and $V_3$, respectively. The relation between $\lambda$ and threshold wavelengths are $\lambda_1 < \lambda$, $\lambda_2 > \lambda$ and $\lambda_3 \gg \lambda$. The correct option is :
An ideal Zener diode with breakdown voltage of $-3\text{ V}$ is reverse biased with a negative input voltage $V_i = -5\text{ V}$. The magnitude of voltage difference between points B and A is :
- Consider the following nuclear reaction :
$$^{238}\text{U} \rightarrow ^{234}\text{Th} + ^{4}\text{He}$$
Take masses of $^{238}\text{U}$, $^{234}\text{Th}$ and $^{4}\text{He}$ as $238.050\text{ u}$, $234.043\text{ u}$ and $4.003\text{ u}$, respectively. The Q value for the reaction, in keV, is :
[Given : $1\text{ u} = 931.5\text{ MeV c}^{-2}$] - A photon and an electron, each of $20\text{ eV}$ energy, move in free space. The ratio of linear momentum of electron $p_e$ to that of photon $p_{ph}$, $\frac{p_e}{p_{ph}}$ is :
(Take speed of light $= 3 \times 10^8\text{ ms}^{-1}$, charge of electron $= -1.6 \times 10^{-19}\text{ C}$ and mass of electron $= 9 \times 10^{-31}\text{ kg}$) - Consider that an electron is revolving in an excited state of Hydrogen atom with velocity $\sqrt{25.6} \times 10^5\text{ ms}^{-1}$. The radius of the orbit is $x \times 10^{-9}\text{ m}$. The value of $x$ is :
[Take the mass of electron to be $9 \times 10^{-31}\text{ kg}$, charge of electron $= 1.6 \times 10^{-19}\text{ C}$ and $\frac{1}{4\pi\epsilon_0} = 9 \times 10^9\text{ N m}^2\text{ C}^{-2}$] - In Geiger-Marsden experiment, the number of scattered $\alpha$-particles $N(\theta)$ is plotted as a function of scattering angle $\theta$. Which of the following options represents the correct plot ?
- A beam of light falls on a metal surface such that photo-electrons are generated. If power of the light source starts to decrease linearly with time $t$, then variation of the photocurrent $I$ and magnitude of the stopping potential $|V|$ with time is best represented by :
Important Questions from Modern Physics
Assuming in forward bias condition there is a voltage drop of $0.7 \text{ V}$ across a silicon diode, the current through diode $D_1$ in the circuit is ________$\text{mA}$.
(Assume all diodes in the given circuit are identical)
- The ratio of de Broglie wavelength of a deutron with kinetic energy $E$ to that of an alpha particle with kinetic energy $2E$, is $n : 1$. The value of $n$ is ________.
(Assume mass of proton = mass of neutron) : - The wave numbers of three spectral lines of H atom are considered. Identify the set of spectral lines belonging to Balmer series.
(R = Rydberg constant) - 7.9 MeV $\alpha$-particle scatters from a target material of atomic number 79. From the given data the estimated diameter of nuclei of the target material is (approximately) _________ m.
$\left[ \frac{1}{4\pi\epsilon_0} = 9 \times 10^9 \text{ Nm}^2/\text{C}^2 \text{ and electron charge} = 1.6 \times 10^{-19} \text{ C} \right]$ Find the correct combination of A, B, C and D inputs which can cause the LED to glow.
