The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass per unit volume. The symbol most often used for density is ρ (the lower case Greek letter rho), although the Latin letter D can also be used. Mathematically, density is defined as mass divided by volume:
ρ
=
m
V
{\displaystyle \rho ={\frac {m}{V}}}
where ρ is the density, m is the mass, and V is the volume. In some cases (for instance, in the United States oil and gas industry), density is loosely defined as its weight per unit volume, although this is scientifically inaccurate – this quantity is more specifically called specific weight.
For a pure substance the density has the same numerical value as its mass concentration.
Different materials usually have different densities, and density may be relevant to buoyancy, purity and packaging. Osmium and iridium are the densest known elements at standard conditions for temperature and pressure.
To simplify comparisons of density across different systems of units, it is sometimes replaced by the dimensionless quantity "relative density" or "specific gravity", i.e. the ratio of the density of the material to that of a standard material, usually water. Thus a relative density less than one relative to water means that the substance floats in water.
The density of a material varies with temperature and pressure. This variation is typically small for solids and liquids but much greater for gases. Increasing the pressure on an object decreases the volume of the object and thus increases its density. Increasing the temperature of a substance (with a few exceptions) decreases its density by increasing its volume. In most materials, heating the bottom of a fluid results in convection of the heat from the bottom to the top, due to the decrease in the density of the heated fluid. This causes it to rise relative to more dense unheated material.
The reciprocal of the density of a substance is occasionally called its specific volume, a term sometimes used in thermodynamics. Density is an intensive property in that increasing the amount of a substance does not increase its density; rather it increases its mass.
For a flat universe, density parameter Ωuniverse=1. How does negative signage of a constituent density parameter, such as that of curvature index Ωk, which can be 0,1,-1 affect the signage of Ωuniverse? If Ωk were to be converted to its energy density, which is much less than the energy density...
Is it correct that:
density = [ 0.09 * (density of NaCl) ] + [0.91 * (density of water) ]
volume = [ (volume of water) + ( { (volume of NaCl) / 48} * 9) ]
Thank you so much for any advice.
We've been talking in another thread about supermassive black holes. That has me thinking about really, really big BH's - so large that the spacetime curvature and evolution of the universe matters.
Let's start by defining the density of a black hole as its mass divided by the volume enclosed...
Good evening,
I'm running into some trouble with this problem, and I have a hint as to why, but I'm not completely sure. Please see the steps below for context.
I've been able to set up the proper equation representing the density as a function of distance from the center which looks like this...
Ok, let's compare two cubes of lead. First lead cube weigh 6078 grams and its area is 3376 cm. The second lead cube is smaller and lighter at 5216 grams and 2713 cm area. The density of the first lead cube which is the bigger and heavier lead (by dividing its weight with the area) is 1.800 g/cm2...
I am not sure if my report is complex enough as it should be at the undergraduate level preferably based on the requirements for it and it feels like it's all over the place as of now.
m * g = mAl * g
V * ρ * g = VAl * ρAl * g
V * ρ * g = V * ρAl * g
ρ = ρAl
this does not work at all, because the upper ball must have a density smaller than that of seawater 1200kg/m3 or not?
I'm trying to figure out how describe the mass of air between the piston face and the und of the tube ( position ##o##) in the acompanying diagram.
At ##t = 0##, the mass of air in the tube is ##M_o##, and the system is static with tube length ##l##. The ##x## coordinate describes how far...
Can someone please help derive the relations below from first principles?
Also does someone please know what happens when ##ρ_{object} = p_{fluid}##?
Many thanks!
I want to find the cumulative mass m(r) of a mass disk. I have the mass density in terms of r, it is an exponential function:
ρ(r)=ρ0*e^(-r/h)
A double integral in polar coordinates should do, but im not sure about the solution I get.
Its form is:
(n2-1)/(n2+2)=(4π/3)Nam
There is one simple problem with it. Rearrange the left side and you get:
(n2+2-3)/(n2+2)=(4π/3)Nam
1-(3/(n2+2))=(4π/3)Nam
As you see, the left side cannot reach unity for arbitrarily large n2.
But there is no reason why N cannot be arbitrarily large!
How...
In the frame of the patch ##-(1/\rho) \nabla p = - \nabla \phi##, and putting ##\nabla p = (\partial p/\partial \rho) \nabla \rho = c_s^2 \nabla \rho## and taking the ##z## component gives\begin{align*}
-\frac{c_s^2}{\rho} \frac{\partial \rho}{\partial z} = -c_s^2...
The sphere floats on water so we should have: ##F_b=F_g##
The buoyant force is equal to the weight of the displaced fluid, so : ##\rho _wV_wg=\rho _sV_sg##
(w: water, s: sphere)
From last equation we have : ##V_w=\frac {\rho _s}{\rho _w} V_s \rightarrow V_w=5 V_s ##
The volume of displaced...
The starting point is the identity
$$\left(\frac{\partial u}{\partial T}\right)_n = T\left(\frac{\partial s}{\partial T}\right)_n.$$
I then try to proceed as follows:
Integrating both with respect to ##T## after dividing through by ##T##, we find
$$ \int_0^T \left(\frac{\partial s}{\partial...
I am studying a 2D material using tight binding. I calculated density of states using this method. Can I also calculate partial density of states using tight binding?
Is there a way to independently determine the proportion of dark energy density to total energy density of the universe apart from using 1 -(Ωmatter+Ωdark matter )?
The correct answer to this problem is: ##\sigma = \varepsilon_0E\frac{\varepsilon-1}{\varepsilon}##
Here is my attempt to solve it, please tell me what is my mistake?
##E_{in} = E_{out} - E_{ind}##
##E_{ind} = E_{out} - E_{in}##
##E_{in} = \frac{E_{out}}{\varepsilon}##
##E_{ind} = E_{out} -...
Hello everyone!
I'm trying to replicate phonon density of states (PHDOS) diagrams for some solids using Quantum Espresso. The usual way I do it is the following one:
scf calculation at minima (pw.x)
Calculation of dynamical matrix in reciprocal space with nq=1 or 2 (ph.x)
Calculation of...
$$n = \sqrt{n_x^2 + n_y^2 +n_z^2}$$
$$E = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$$
$$n = \sqrt{ \frac{2mL^2E}{\pi^2 \hbar^2} }$$
This is all given by the textbook.
It's even as friendly as to say
$$\text{differential number of states in dE} = \frac{1}{8}4 \pi n^2 dn$$
$$D(E) = \frac{...
Hello, I am going over the derivation for two-electron density. I am having a hard time understanding how the second term in 2.11a seen below is derived. I know this term must eliminate the i=j products but can't seem to understand how. Thanks for the help.
I am confused about how the electric field changes in this problem - is E' = E/Ke=E/2? Is E = V/d a correct usage?
When I solve it this way, the answer is incorrect:
change in energy density = (1/2)ε(E'2- E2) = (1/2)ε(E2/4 - E2) = (1/2)ε(-3/4)(V/2d)2.
What am I doing wrong? Thanks.
Here's my attempt at a solution, but when I plug it in, it gives me a power ten error. I don't really understand what I'm doing wrong here. I think all my variables are in the correct units and it asks for my answer to be in μC/m2. Any help is much appreciated.
Okay so I am a little confused as to where I made a mistake. I couldn't figure out how to program Latex into this website but I attached a file with the work I did and an explanation of my thought process along the way.
Schrodinger’s original interpretation of the wavefunction was that it represented a smeared out charge density however this was replaced with Max Born’s probability interpretation. The issue was from what I understand that a charge density would repel and have self interactions as all the charge...
Hi all, in this question i was asked to find the percentage change in the density, my approach was as following, first i find the change in volume due to putting the gas into the other vessel as:
$$
P_{1}V_{1}=P_{2}V_{2}\;\; → \;\;V_{2}=\frac{P_{1}}{P_{2}}V_{1}
$$
now i use
$$...
If we throw empty but sealed glass bottle in water it will float with 60% of it's volume above the surface of water.
What is average density of floating bottle and what is it's volume? Mass of bottle is 0.4 kg(mass of air inside of bottle is irelevant).
It's easy problem but I can't get right...
For part a:
I know that linear charge density is the amount of charge per unit length, and we are given the volume charge density. Since we are given the volume, we can obtain the length by multiplying the volume by the cross sectional area, so C/m^3 * m^2 = C/m. The cross sectional area of a...
Hello
Please help me. I'm not a chemistry student and I don't have a chemistry-related course, so please explain in a very simple way. Thank you.
I have a composite composition that I only have the weight percentage of atoms and I need to calculate the density so that I can check the properties...
When arriving at the standard model of cosmology, i.e. the exapnding universe, we assume based on experirmental data that the cosmos is homogenous on large enough scales.
But when we go back in time, when the galaxies are beginning to form, we note that because of the growth of density...
Surface acceleration is proportional to density and radius of planet (as 2 powers of R cancel with the volume)
g(moon)/g(earth) = density(moon)*radius(moon)/density (earth)*radius(earth) = (1/4)*density(moon)/density(earth)
I'm trying to find the purification of this density matrix
$$\rho=\cos^2\theta \ket{0}\bra{0} + \frac{\sin^2\theta}{2} \left(\ket{1}\bra{1} + \ket{2}\bra{2} \right)
$$
So I think the state (the purification) we're looking for is such Psi that
$$
\ket{\Psi}\bra{\Psi}=\rho
$$
But I'm not...
Sakurai, in ##\S## 5.7.3 Constant Perturbation mentions that the transition rate can be written in both ways:
$$w_{i \to [n]} = \frac{2 \pi}{\hbar} |V_{ni}|^2 \rho(E_n)$$
and
$$w_{i \to n} = \frac{2 \pi}{\hbar} |V_{ni}|^2 \delta(E_n - E_i)$$
where it must be understood that this expression is...
From Rand Lectures on Light, we have, in the interaction picture, the equation of motion of the reduced density matrix:
$$i \hbar \rho \dot_A (t) = Tr_B[V(t), \rho_{AB}(t)] = \Sigma_b \langle \phi_b | V \rho_{AB} -\rho_{AB} V | \phi_b \rangle = \Sigma_b \phi_b | \langle V \rho_{AB} | \phi_b...
It is a 1D Tonk gas consisting of ##N## particles lined up on the interval ##L##. The particles themselves have the length ##a##. Between two particles there is a gap of length ##y_i##. ##L_f## is the free length, i.e. ##L_f=L-Na##.
I have now received the following tip:
Determine the...
My answer given below seems incomplete.
Since warm air causes the air to expand in volume, so its density becomes less as compared to the colder air at the top of the room. After this, I generally find all books saying the less dense air rises and more dense air from top comes down and...
Is there a way to obtain equation 9.42 (I is current, j is current density, and sigma is conductivity) in the following image (from Modern Electrodynamics by Andrew Zangwill, the part on electromotive force) besides using V=IR and substituting the line integral of j/conductivity for V? The...
I developed three arguments to answer this question. Argument no 2 seems to be wrong, but I cant figure out why. I know one/more of my arguments are flawed. Please be kind to help me figure this out.
Argument 1) Since they have same charge on them, the ##E## between them must be same. The one...
Hello! I am using a mixture density network (MDN) to make some predictions. My model is very simple with one hidden layer only with 10 nodes (the details of the network shouldn't matter for my question but I can provide more if needed). Also my MDN has only one gaussian component which basically...
Part (a) was simple, after applying
$$Q=\int_{\mathbb{R}^3}^{}\rho \, d^3\mathbf{r}$$
I found that the total charge of the configuration was zero.
Part (b) is where the difficulties arise for me. I applied
$$V(\mathbf{r})=\frac{1}{4\pi \epsilon _0}\int_{\Gamma }^{}\frac{\rho...
I am refreshing on this; ..after a long time...
Note that i do not have the solution to this problem.
I will start with part (a).
##f(u)= 3u-\dfrac{3u^2}{2k}## with limits ##0≤u≤k##
it follows that,
##3k - \dfrac{3k}{2}=1##
##\dfrac{3k}{2}=1##
##k=\dfrac {2}{3}##
For part (b)...
hi, i have seen lagrangian density for spin 0 , spin 1/2, spin 1 , but i am not getting from where these langrangian densities comes in at a first place. kindly give me the hint.
thanks
The vapour density of N204 at certain temperature is 30. Calculate the percentage of dissociation of N204 at this temperature. N2O4(g)⇌2NO2(g)?
I am unable to understand the concept behind vapour density of the mixture.
Currently, I understand that
2 x vapour density=molar mass.
Vapour density =...
a.)
The scale length of the disk is the length over which the surface density of stars decreases by a factor of e. In this case, the surface density decreases by a factor of 10 over a distance of 9 kpc, so the scale length is 9 kpc. The surface density of stars at a radius of r from the center...
What is the flux density of salt in a horizontal tube 10 cm in length connecting seawater (salinity = 30 g/l) to a tank of freshwater (salinity ~ 0) assuming no advection occurs?